THE  LIBRARY 

OF 

THE  UNIVERSITY 

OF  CALIFORNIA 

Education 

IN  MEMORY  OF 

Professor 
George  D.  Louderback 
1874-1957 


APPLETONS'  STANDARD  ARITHMETICS 


NUMBERS    APPLIED 


A   COMPLETE   ARITHMETIC 


BY 


Andrew  J.  Rickoff,  A.M.,  LL.D. 


NEW  YORK,  BOSTON,  AND  CHICAGO 

D.    APPLETON    AND     COMPANY 
1887 


Copyright,  1886, 
By  D.  APPLETON  AND  COMPANY. 


£duo.  Lib. 


GIFT 


PEEFACE. 


Tms  work  is  not  the  result  of  any  ambition  on  the  part  of  the  publishers 
to  add  another  title  to  their  already  long  list  of  text-books,  but  of  a  desire 
to  meet  a  wide-spread  and  growing  demand  for  a  treatise  on  arithmetic 
adapted  to  the  objective  methods  of  instruction  now  so  common  in  all  edu- 
cational institutions  which  have  been  reached  directly  or  indirectly  by  the 
influence  of  normal  schools,  teachers'  institutes,  etc. 

In  its  preparation  the  author  has  kept  steadily  in  view  these  two 
thoughts :  (1)  That  words  are  useless  in  the  ratio  that  they  fail  to  call  up 
in  the  mind  vivid  images  of  the  things  signified.  Hence  the  aim  to  vitalize 
the  relation  of  words  and  things  by  the  aid  of  the  best  practicable  illustra- 
tions at  every  point ;  and  (2)  That,  to  the  learner,  the  operations  of  arith- 
metic are  apt  to  be  but  manipulations  of  figures  after  prescribed  models, 
unless  he  realizes  the  fact  that  they  are  representative  of  processes  that 
may  be  applied  to  material  objects. 

The  book  is  intended  to  be  put  into  the  hands  of  the  learner  as  soon  as 
he  has  completed  a  course  in  primary  arithmetic  ;  but  it  would  be  well  for 
him  to  begin  the  study  of  it  with  the  first  chapter,  that  he  may  get  a  better 
technical  knowledge  of  the  fundamental  rules  and  their  relations  to  each 
other,  and  that  he  may  become  rapid  and  reliable  in  computations  involving 
integers  before  he  takes  up  the  more  complicated  subject  of  fractions. 

Great  care  has  been  taken  to  adapt  the  work  as  far  as  possible  to  the 
needs  of  the  great  number  of  children  who  are  withdrawn  from  school 
before  a  full  course  in  arithmetic  can  be  completed.  With  this  object  in 
view,  the  more  useful  business  applications  of  elementary  principles  are 
made  as  soon  as  they  are  learned.  Thus,  familiar  measures  are  introduced 
before  reduction  is  mentioned  ;  federal  money  before  decimals ;  many  prac- 
tical measurements  before  mensuration ;  and  questions  even  in  percentage 
and  interest  are  to  be  met  with  before  those  subjects  are  reached  in  due 
course.     The  conditions  of  these  problems  are  so  presented  as  to  be  within 

965 


iv  PREFACE. 

the  easy  understanding  of  the  pupil,  while  their  solution  requires  only  such 
arithmetical  operations  as  he  has  already  learned. 
Attention  is  respectfully  called 

1.  To  the  simple  treatment  of  the  decimal  system  of  notation,  and  the 
great  number  of  exercises  intended  to  familiarize  the  pupil  with  the  facili- 
ties for  calculation  which  it  affords. 

2.  To  the  multiplicity  of  short  exercises  that  can  be  performed  without 
the  aid  of  the  pencil,  or  that  require  but  few  figures  in  their  solution. 
Longer  ones  are  not  wanting  to  test  the  perseverance  of  the  pupil. 

3.  To  the  directions  to  the  pupil,  having  in  view  the  formation  of  right 
habits  of  computation.  The  ''making  up  "  method  of  subtraction,  and  the 
so-called  "continental"  method  in  division,  though  not  obtrusively  pre- 
sented, are  worthy  of  the  attention  of  teachers.  The  latter  furnishes  an 
excellent  mental  exercise. 

4.  To  the  suggestions  for  original  problems,  now  so  commonly  resorted 
to  by  the  best  teachers  to  stimulate  the  interest  of  their  pupils,  and  to  give 
them  a  better  understanding  of  the  subjects  to  which  they  relate.  Their 
usefulness  as  short  practical  exercises  in  penmanship,  spelling,  and  com- 
position, will  be  appreciated  by  all. 

5.  To  the  simple  and  direct  methods  of  treating  the  fundamental  rules, 
common  and  decimal  fractions,  percentage,  interest,  proportion,  square 
and  cube  roots,  the  problems  of  mensuration,  etc. 

6.  To  the  rigorous  adherence  throughout  the  work  to  the  inductive 
methods  of  instruction. 

The  number  and  variety  of  exercises  and  problems  in  this  work  are  so 
great  as  to  supersede  any  necessity  for  a  supplementary  book  of  exercises. 

It  is  earnestly  recommended  that  the  pictured  illustrations  may  be 
regarded  as  merely  suggestive  of  the  objective  demonstrations  which  the 
student  should  be  encouraged  to  get  up  for  himself.  As  far  as  possible, 
let  the  learner  furnish  all  the  apparatus  needed.  While  he  is  engaged  in 
preparing  it,  the  principles  to  be  illustrated  will  present  themselves  to  his 
mind  more  forcibly  than  in  the  repetition  of  definitions  and  rules  in  which 
he  can  take  but  slight  interest  till  he  appreciates  their  significance.  If  this 
course  be  taken,  the  pupil  will,  in  most  cases,  be  able  to  make  out  his  own 
analyses.  These  may  be  crude  at  first,  but  they  will  be  the  better  for  being 
his  own.  Observation  and  experience  will  guide  to  better  forms.  Such  a 
method  will  give  him  a  mastery  of  the  subject,  develop  mental  power,  and 
cultivate  a  taste  for  independent  investigation. 

New  Yokk  City,  May  15,  1886. 


CONTENTS 


PAGE 

Notation  and  Numeration  ...  3 
Roman  Notation 18 

Addition    ....  * 19 

Blackboard  Exercises  ....  26 
Original  Problems 34 

Subtraction 35 

Miscellaneous  Examples  ...  4*7 
Original  Problems 50 

Multiplication    .     .     .     .     .     .     .     51 

Familiar  Measures 65 

Original  Problems 70 

Division VI 

Original  Problems 93 

Self-Testing  Exercises   ....     96 

Miscellaneous  Examples  .     .     .     .     97 

United  States  Money 107 

Making  Cbange 121 

Factors  and  Divisors 123 

Factors  and  Multiples    .     .     .     .131 

Common  Fractions 135 

Introductory  Exercises  .     .     .     .135 

Reduction    .     .     , 140 

Addition 143 

Subtraction 145 

Multiplication 149 

Division 155 

Aliquot  Parts 164 

Miscellaneous  Examples     .     .     .166 

Decimal  Fractions 173 

Introductory  Exercises  .  ,  .  .173 
Addition      . 180 


Decimal  Fractions  {continued). 

Subtraction      .     .     .     .     .     .     .181 

Multiplication  .......  182 

Division 184 

Reduction 186 

Miscellaneous  Examples     .     .     .191 

Bills  and  Accounts 194 

Original  Problems     .....  200 

Measures 201 

Of  Extension 201 

Of  Capacity 205 

Of  Weight  .     .     .     .  •  .     .     .     .207 
Of  Value     ........  210 

Of  Arcs  and  Angles 212 

Of  Time 213 

Miscellaneous  Problems     .     .     .  214 

Compound  Denominate  Numbera    .215 

Reduction .218 

Addition 224 

Subtraction 226 

Multiplication 227 

Division 228 

Longitude  and  Time 236 

Application  and  Review     .     .     .  237 
Miscellaneous  Problems     .     .     .  242 

The  Metric  System  of  Weights  and 

Measures 249 

Reduction 254 

Square  Measure 256 

Cube  Measure 258 

Wood  Measure     ......  259 


VI 


CONTENTS. 


PAGE 

Practical  Measurements  ....  262 

Lumber 262 

Masonry  and  Brickwork     .     .     .  263 

Flooring 264 

Plastering 265 

Painting  and  Kalsomining .     .     .  265 

Paper  Hanging 266 

Carpeting 266 

Paving 267 

Bins,  Tanks,  and  Cisterns  .     .     .  268 
Estimating  the  Weight  of  Hay  in 

a  Mow 268 

Miscellaneous  Problems     .     .     .  269 
Original  Problems 270 

Percentage 271 

Loss  and  Gain 277 

Trade  Discount 285 

Insurance 286 

Commission  and  Brokerage     .     .  288 

Stocks 290 

Taxes 292 

Miscellaneous  Problems     .     .     .  295 
Original  Problems 800 

Interest .     .  301 

Present  Worth 316 

Exact  Interest 317 

Common  Business  Method .     .     .  318 

Bank  Discount 319 

Promissory  Notes 321 

Partial  Payments 323 

United    States  Rule  for  Partial 

Payments 324 

Mercantile  Rule  for  Partial  Pay- 
ments  326 

Annual  Interest 327 

Miscellaneous  Problems     .     .     .  328 
Compound  Interest 331 

Equation  of  Payments  .  .  .  .333 
Debit  and  Credit  Accounts  .  .  340 
Original  Problems     .....  342 


PAGE 

Proportion 343 

Inverse  Proportion 345 

Ratio  and  Proportion  ....  347 
Compound  Proportion  ....  352 
Miscellaneous  Problems     .     .     .  355 

Squares  and  Cubes 357 

Square  Root 363 

Cube  Root £68 

Extraction  of  Roots  by  Factoring.  371 
Geometric  Solution  of  the  Prob- 
lem of  Square^  Root   ....  372 
Geometric  Solution  of  the  Prob- 
lem of  Cube  Root 373 

Applications  of  Square  and  Cube 

Root 374 

Right-angled  Triangles  ....  375 

Mensuration 377 

Of  Plane  Surfaces 377 

Of  surfaces  of  Prisms,  etc.  .  .  386 
Of  Volume  or  Contents  of  Solids.  390 

Duodecimals 392 

Original  Problems 394 

Exchange 395 

Domestic  Exchange 395 

Foreign  Exchange 400 

Duties  or  Customs 403 

Bonds 404 

APPENDIX. 
Testing  the  Accuracy  of  Addition, 
Subtraction,     Multiplication, 

and  Division 407 

Casting  out  Nines 407 

Greatest  Common  Divisor     .     .    .  409 

Circulating  Decimals 41 0 

Progression 411 

Arithmetical  Progression  .     .     .411 

Geometrical  Progression    .     .     .413 

Values  of  Foreign  Coins  .    .     .     .415 

Legal  Rates  of  Interest  .     .     .     .416 


^PPLETO^S' 

Standard  ^ritlmietic. 


CHAPTER  I. 

NOTATION    AND    NUMERATION. 

The  Writing  and  Reading  of  Numbers. 

NUMBER.  OBJECTS.  NUMBER. 


• 

One 

1 

•  • 

•  • 

• 
• 

Six 

6 

•  • 

Two 

2 

:•: 

• 
• 

Seven 

7 

• 
•  • 

Three 

3 

•  #  •  • 

•  •  e  • 

Eight 

8 

•  • 

•  • 

Four 

4 

v  •• 

•  •  •  • 

Nine 

9 

:•: 

Five 

5 

V  V 

•  •  •  • 

Ten 

•• 

(.  The  signs  1,  2, 3, 4, 5,  6, 7,  8, 9,  are  called  the  nine  digits; 
because  first  used  to  represent  a  number  of  fingers. 

The  word  digit  is  sometimes  used  for  the  word  finger. 


4  STANDARD  ARITHMETIC. 

The  following  are  the  written  forms  of  the  digits  : 

/.   z.  <3.   a.  &  6,  y.  fi   <?, 


Tens  and  Units. 

2.  When  we  count  more  than  nine  we  begin  to  count  by  tens 
and  ones.  After  nine  we  say  ten,  then  eleven,  which  means  ten 
and  one,  twelve  (ten  and  two),  thirteen  (ten  and  three),  fourteen 
(ten  and  four),  etc.,  to  nineteen  (ten  and  nine).  Then  we  come 
to  twenty,  which  means  two  tens,  twenty-one  (two  tens  and  one), 
etc.,  after  twenty-nine  we  haye  thirty,  forty,  fifty,  etc. 

Counting. — Count  the  balls  of  the  numeral  frame  or  other 
objects  from  ten  to  ninety-nine,  as  follows  ■ 

ten  and  one  two  tens  and  one  three  tens  and  one 

ten  and  two  two  tens  and  two  three  tens  and  two 

ten  and  three  two  tens  and  three  three  tens  and  three 

etc.,  to  etc.,  to  etc.,  to 

ten  and  nine  two  tens  and  nine  three  tens  and  nine 

two  tens  three  tens  four  tens,  etc. 

Writing. — We  may  write  these  numbers  by  using  the  digits 
1,  2,  3,  etc.,  instead  of  the  words  one,  two,  three ;  thus  : 

1  ten  and  1  2  tens  and  1  3  tens  and  1 

1  ten  and  2  2  tens  and  2  3  tens  and  2 

1  ten  and  3  2  tens  and  3  3  tens  and  3 

etc.,  to  etc.,  to  etc.,  to 

1  ten  and  9  2  tens  and  9  3  tens  and  9 

2  tens  3  tens  4  tens 

and  so  on  to  9  tens  and  9. 

3.  But  the  writing  of  numbers  is  still  further  shortened  by 
omitting  the  words  ten  and,  or  tens  and ;  thus,  for  1  ten  and  1 
we  write  11 ;  for  1  ten  and  2  we  write  12 ;  for  2  tens  and  1  we 
write  21,  and  so  on  up  to  99  (9  tens  and  9).  Thus  we  can  tell 
whether  a  digit  stands  for  ones  or  tens  by  the  place  it  occupies. 
If  it  represents  ones,  it  has  the  first  place  at  the  right ;  if  tens,  it 
occupies  the  next  on  the  left. 


NO  TA  TION  A  ND  NUMERA  TION.  5 

4B  The  right-hand  place  is  called  the  units'  or  ones'  place. 
The  next  place  to  the  left  of  the  units  is  the  tens'  place. 

Note. — We  use  the  word  unit  for  the  word  one,  and  units  for  ones.     See  Art.  12. 

If,  as  in  1  ten,  2  tens,  etc.,  there  are  no  ones  or  units  to  be 
expressed,  we  write  the  figure  0  in  the  units'  place ;  thus,  10,  20. 

5.  The  figure  0  does  not  express  any  number,  but  it  is  used 
to  fill  vacant  places,  as  in  the  case  above.  It  is  called  cipher 
or  zero.     It  is  a  figure,  but  not  a  digit. 

6.  This  is  the  decimal  or  tens'  system  of  counting  and  of 
writing  numbers. 

Note. — For  a  simple  II- 
r -a     -     ;     ^  lustration  of  this    system, 

imagine  a  boy  counting 
sticks.  He  counts  ten  and 
ties  them  together ;  then 
ten  more,  and  so  on  till  he 
has  seven  bundles  and  five 
sticks  over.  Then,  writing 
15,  he  shows  that  he  has 
seventy-five  sticks,  the  1 
being  in  tens,  and  the  5  in 
units'  place. 

7.  Between  twelve  and  twenty  we  call  the  ten  "teen"  which 
means  "and  ten";  as,  fourteen,  that  is  four  and  ten.  From 
nineteen  to  ninety-nine  we  call  the  ten  "ty,"  which  means  "times 
ten";  as,  sixty,  or  six  times  ten;  seventy-five,  or  seven  times  ten 
and  five  units. 


EXERCISES     IN     WRITING     AND     READING     NUMBERS. 

Note. — Pupils  should  follow  the  forms  of  the  digits  given  at  the  top  of  the 
preceding  page,  or  other  good  copy.  Let  him  here  lay  the  foundation  of  neatness 
and  accuracy  in  the  writing  and  use  of  figures. 

l.  Write  very  neatly,  in  figures,  the  numbers  from  one  to  nine  ; 
from  thirty  to  thirty-nine  ;  forty  to  forty-nine  ;  nineteen  to  ten  ; 
sixty  to  sixty-nine  ;  ninety  to  ninety-nine  ;  eighty-nine  to  seventy  ; 
fifty-nine  to  fifty  ;  thirty-four  to  fifty-six. 


6  STANDARD  ARITHMETIC. 

2.  Write  in  words  :  75,  43,  51,  98,  29,  83,  11,  3%  64,  49,  17, 

56,  19,  68,  31,  77,  99,  10,  24,  48. 

Note. — The  pupils  may  be  required  to  show  a  number  of  jack-straws  or  of 
other  objects  equal  to  the  numbers  expressed.  They  should  be  arranged  appro- 
priately in  tens  and  units. 

3.  How  many  tens  are  in  twenty-eight  ?  In  sixty-two,  fifty- 
six,  ninety-five,  eighty,  seventy-one,  forty-eight,  sixty-five,  thirty- 
three,  fifty-one,  etc.  ? 

4.  Write  the  following  numbers  in  a  column,  and  opposite 
each  one  express  the  same  value  in  words  :  92,  65,  38,  71,  40,  83, 
14,  54,  17,  26,  90,  12,  83,  24,  75,  36. 

5.  The  number  39  is  expressed  by  two  digits.  What  does  the 
9  stand  for  ?  The  three  ?  Which  one,  as  it  stands  here,  expresses 
the  greater  value  ?    Why  ? 

6.  Answer  the  same  questions  in  regard  to  89,  48,  56,  35,  67, 
22.     Illustrate  by  objects. 

Note. — In  English  the  digit  expressing  the  tens  is  generally  read  first;  as, 
f orty-cight ;  but  in  the  German  language  they  say:  eight  and  forty.-  Sometimes 
we  hear  the  same  in  English. 

7.  Read  the  following  numbers  in  the  German  way  :  27  (seven 
and  twenty),  36,  58,  67,  89,  38,  45.  If  the  places  of  these  digits 
were  exchanged,  would  the  numbers  thus  expressed  be  larger  or 
smaller  ?    Why  ?    Illustrate  by  objects. 

8.  What  is  the  greatest  number  that  can  be  expressed  by  two 
figures  ?    What  is  the  smallest  ? 

9.  Write  in  figures  :  Seventeen,  thirteen,  fifteen,  twenty-eight, 
ninety-five,  forty-two,  eighty-three,  thirty-four,  sixty-nine,  seventy- 
seven,  eighteen,  fifty-one,  sixty-seven,  forty-eight,  ninety-five, 
eighty-eight,  thirty,  sixty. 

10.  Read  the  following.  (May  be  copied,  or  written  at  dicta- 
tion, and  then  read.)  10,  19,  13,  18,  12,  17,  11,  16,  14,  20,  23, 
21,  25,  27,  29,  22,  24,  26,  28,  32,  36,  31,  35,  33,  38,  37,  49,  44, 
48.  41,  47,  50,  56,  52,  63,  75,  84,  95,  58,  67,  78,  89,  91,  65,  85, 
96,  72,  15,  30,  35,  39,  42,  55. 


NOTATION  AND  NUMERATION.  7 

11.  The  pupil  may  copy  the  following  numbers  : 

56     34     67    23     11     89     78     45     98     10     87    54     32     43     21 

65     76     17    26    49     61     88     39    58     72     99     30    27    83    57 

Note. — Care  should  be  taken  that  each  number  be  recognized  as  a  whole,  that 
the  pupil  may  not  copy  figures  merely.  He  should  recognize  56  as  fifty-six,  and  not 
as  the  figures  5  and  6. 

12.  Take  in  at  a  glance  as  many  numbers  as  you  can,  and 
repeat  them,  looking  off  the  book  : 

16        23         62        43        50        84        31         74        39        20 
78        47        39         91         98        67        17        58        46        55 

13.  The  teacher  may  dictate  two  or  more  numbers  at  a  time 
from  exercises  10  and  11. 

14.  Write  the  number  of  jack-straws  represented  in  each  group 
below.     The  bundles  are  of  ten  each. 


Fiv%> 


Suggestion. — Let  the  pupils  make  original  notation  exercises  similar  to  the 
above,  arranging  the  objects  in  groups,  and  noting  the  number  both  in  words  and 
figures. 

8.  Hundreds. — If  we  count  one  more  than  ninety-nine  we 
shall  have  nine  tens  and  ten  ones,  or  ten  tens.  Ten  tens  make 
one  hundred.  To  express  one  hundred  in  figures  we  write  1  in 
the  third  place,  thus,  100,  filling  the  places  of  tens  and  units  with 
ciphers.  The  1  now  stands  for  one  hundred.  A  digit  in  the 
third  place  from  the  right  stands  for  hundreds,  and  hence  we 
write : 

100  (one  hundred),  400  (four  hundred),       700  (seven  hundred), 

200  (two  hundred),  500  (five  hundred),        800  (eight  hundred), 

300  (three  hundred),       600  (six  hundred),  900  (nine*  hundred). 

If  with  the  hundreds  we  have  to  write  any  number  of  tens, 
as  three  hundreds  and  seven  tens,  we  place  the  digit  representing 


8 


STANDARD  ARITHMETIC. 


the  tens  in  the  tens'  place  ;  thus,  370,  read  (3  hundreds  7  tens), 
three  hundred  seventy. 

Again,  if  with  the  hundreds  we  have  to  write  any  number  of 
ones,  as  three  hun- 
dreds and  five  ones, 
we  place  the  figure  re- 
presenting the  ones  in 
the  ones'  place  ;  thus, 
305. 

Three    hundreds, 
seven   tens   and  five 
ones  here  represented  are  written  thus  :  375,  and  read  three  hun- 
dred seventy-five. 

We  have  learned,  1st,  that  ten  ones  make  one  ten,  and  ten  tens 
make  one  hundred ;  2d,  that  in  writing  numbers  the  place  on  the 
right  is  the  ones'  place ;  the  next,  the  tens'  place;  and  the  next, 
the  hundreds'  place. 

EXERCISES     IN     WRITING     AND     READING     NUMBERS. 

1.  Express  in  figures  :  One  hundred,  six  hundred,  nine  hun- 
dred, seven  hundred,  four  hundred,  two  hundred,  etc. 

2.  Also,  one  hundred  thirty,  six  hundred  twenty,  five  hun- 
dred eighty,  three  hundred  fifty,  two  hundred  seventy,  etc. 

3.  Also,  one  hundred  sixty-five,  three  hundred  eighty-four, 
nine  hundred  seventy-one,  four  hundred  thirty-three,  etc. 

4.  How  many  hundreds  in  481  ?  How  many  tens  ?  How 
many  ones  ?    How  many  of  each  in  385,  610,  974,  572,  137,  448  ? 

5.  Write  the  following  numbers  in  a  column,  and  opposite 
each  the  same  number  in  words  :  218,  117,  916,  675,  854,  370, 
523,  388,  446,  770,  978,  101,  340,  620,  304. 

6.  Eead,  taking  in  at  one  glance  as  many  numbers  as  possible  : 
100         201         310         404         500         691         700         800         909 
102        204        320      .440         572         673         719         808        910 
Note. — The  foregoing  numbers  may  be  read  in  lines  or  columns,  forward  or 

backward,  as  the  teacher  may  direct. 


NOTATION  AND  NUMERATION. 


9 


7.  Write  in  figures  the  number  of  jack-straws  represented  in 
each  of  these  groups. 


%V».v 


8.  Read  792.  What  does  the  2  stand  for  ?  The  9  ? .  The 
7  ?— Show  what  each  figure  stands  for  in  the  following  numbers  : 
439,  562,  101,  760,  875,  460,  140,  104,  583,  61. 

9.  In  a  number  of  three  places,  which  figure,  is  read  first  ? 
Which  represents  the  highest  order  ?  How  would  you  write  three 
hundred  nine,  having  no  tens  ?  How  would  you  write  7  hundred 
twenty,  having  no  ones  ?  Will  it  do  to  leave  the  place  of  the  ones 
or  tens  vacant  ?     Why  ? 

10.  What  is  the  largest  number  that  can  be  represented  by 
three  figures  ?    What  is  the  smallest  whole  number  ? 

11.  Write  in  figures  :  Three  hundred  fifty,  six  hundred  eighty, 
two  hundred  seventy,  eight  hundred  fifteen,  four  hundred  twenty- 
eight,  nine  hundred  nine,  one  hundred  ninety-six. 

12.  Copy  the  following,  glancing  at  each  number  but  once : 
(Think  of  the  numbers  represented,  not  merely  of  the  figures  to 
be  written.) 


107 

400 

212 

560 

309 

653 

356 

365 

635 

536 

801 

118 

180 

870 

357 

429 

560 

608 

742 

897 

215 

419 

711 

999 

233 

100 

677 

822 

301 

405 

103 

205 

340 

409 

583 

655 

728 

846 

979 

893 

13. 

Note.- 

—The  teacher  ma 

y  also  dictate  two 

or  more 

of  the  i 

foregoii 

lg  num- 

bers  at  once,  thus  quickening  the  attention  of  the  pupils. 

14.  Write  in  regular  order  the  numbers  from  150  to  199  ;  from 
260  to  299  ;  from  307  to  328  ;  from  480  to  499  ;  from  585  to  602 ; 
from  687  to  706  ;  from  791  to  809. 

15.  Write  in  words  the  numbers  from  337  to  345  :  also  from 


10 


STANDARD  ARITHMETIC. 


883  to  890  ;  from  555  to  563  ;  from  98  to  104 ;  from  872  to  883  ; 
from  190  to  205. 


9.  Thousands. — The  greatest  number  we  have  written  thus 

far  is  999,  or  9  hun- 
dreds, 9  tens,  9  ones. 
If  we  count  one 
more,  the  nine  ones 
at  the  right  hand 
will  become  one  ten ; 
and  putting  this  with 

the  nine  tens,  we  have  ten  tens,  or  one  hundred. 

Putting  this  one  hundred  with  the  nine  hundreds,  we  have 

ten  hundreds;  and,  as  we  made 

one  ten  out  of  ten  ones,  and  one 

hundred  out  of  ten  tens,  so  we 

make  one  thousand  out  of   ten 

hundreds.      Thus,    after   adding 

one  stick  to   the  nine  hundred 

and    ninety-nine  shown  on    the 

table  above,  the  result  would  be  as  represented  in  this  picture. 
To  express  one  thousand  in  figures,  we  write  1  in  the  fourth 

place ;  thus,  1000,  filling  the  places  of  hundreds,  tens,  and  units 

with  0's.     The  1  now  stands  for  one  thousand.     A  digit  in  the 

fourth  place  stands  for  thousands,  hence  we  have   2000   (two 

thousand),  etc.      Hundreds,  tens,  and  units,  if  any,  fill  their 

proper  places. 


EXERCISES      IN      READING     AND      WRITING      NUMBERS. 

1.  Read  77,  15,  93,  106,  601,  810,  7080,  9107,  5006,  561,  3091. 

2.  Write  at  dictation  and  read  : 

5783       2100       9009       1706      5430      8071       2360      3902      1003      5701 
6702      4000      3201       4300      5701       7010       8090       9100      1901      7707 

3.  Write  ten  such  numbers  as  you  please,  and  read  them. 


NOTATION  AND  NUMERATION. 


11 


4.  In  the  following  numbers,  how  many  units,  tens,  and  hun- 
dreds are  expressed  by  the  figures  in  those  orders  ? 

2375  1318  2380  3689  7401  8120  9000  7895 

3624  5074  6138  8376  2380  8016  7980  1234 

5.  What  is  the  smallest  whole  number  that  can  be  expressed 
by  four  figures  ?    What  is  the  greatest  ? 

6.  Read; 

6789  9867  5432  4756  8912  3129  7891  4321 

6000  6600  6660  6666  7654  6035  8765  8003 

1098  2076  3054  4032  5010  7123  7009  5273 

9002  9387  8793  1100  ,     4002  7628  9347  6102 

7.  Read  the  foregoing  columns  downward  and  upward,  and 
the  lines  from  right  to  left  and  left  to  right.  They  may  also  be 
written  at  dictation  and  read. 

8.  Read  the  numbers  of  Exercise  4,  reading  the  thousands 
and  hundreds  together  as  hundreds.  Thus,  23 75= twenty- three 
hundred  seventy-five. 


9.  Copy  the  numbers  of  Exercise  6.  No  copying  of  single 
figures  should  be  allowed ;  the  number  should  be  recognized  and 
written  as  a  whole. 

10.  Which  figure  in  a  number  of  four  places  is  read  first  ? 
Which  represents  the  highest  order  ?    Which  the  lowest  ? 

11.  How  many  ciphers  are  needed  in  4  thousand  17  ?    Why  ? 


12 


STANDARD  ARITHMETIC. 


What  difference  is  there  between  the  written  forms  468  and  4608  ? 
Between  375  and  3705  ? 

12.  Is  there  any  difference  between  the  numbers  indicated  by 
46  in  468  and  in  4608  ?  Is  the  value  of  8  in  one  number  different 
from  its  value  in  the  other  ?    Why  ? 

13.  What  number  is  expressed  by  the  figure  9  in  7009  ?  In 
7900  ?     In  7090  ? 


Review  Exercises. 


1.  Write  in  columns  the  figures  which  express  the  following 
numbers  : 

four  hundred 
one  hundred 
five  hundred 
two  hundred 
nine  hundred 
six  hundred 
three  hundred 
eight  hundred 


four 

one 

five 

two 

nine 

six 

three 

eight 


four  thousand 
one  thousand 
five  thousand 
two  thousand 
nine  thousand 
six  thousand 
three  thousand 
eight  thousand 


forty 

ten 

fifty 

twenty 

ninety 

sixty 

thirty 

eighty 

2.  How  can  you  make  the  digits  in  your  first  column  express 
tens  ?  (Answer  :  By  annexing  a  cipher.)  How  hundreds  ?  How 
can  you  make  the  digits  in  the  second  column  express  units  ? 
How  hundreds  ?  How  can  you  make  the  digits  in  the  third 
column  express  tens  ?  How  ones  ?  (Answer :  By  erasing  two 
ciphers.)  Make  the  digits  of  the  fourth  column  express  hun- 
dreds ;  also  tens.  Will  it  change  the  value  of  the  digits  to 
place  a  cipher  at  their  left  ? 

3.  Express  in  figures: 
sixteen  one  hun.  seven 


twenty-nine 

fifty-two 

thirty-six 

seventy-eight 

forty-five 

ninety-four 

eighty-three 


three  hun.  eighteen 
five  hun.  twenty-six 
seven  hun.  sixty-four 
eight  hun.  fifty-six 
nine  hun.  thirty-eight 
two  hun.  eighty-nine 
four  hun.  forty-five 


three  thou,  seven  hun.  eight 
six  thou,  one  hun.  twelve 
one  thou,  six  hun.  thirteen 
eight  thou,  two  hun.  twenty 
three  thou,  four  hun.  fourteen 
four  thou,  nine  hun.  ninety 
two  thou,  six  hun.  ten 
five  thou,  three  hun.  thirty 


NOTATION  AND  NUMERATION.  13 

4.  Make  60,  40,  80,  10,  30,  50,  90,  70,  20  larger  by  100 ;  by 
300  ;  by  500  ;  by  400. 

5.  Make  61,  42,  53,  74,  85,  26,  37,  18  larger  by  400 ;  by 
200 ;  by  700. 

6.  Would  it  alter  the  value  of  8  in  81  if  you  were  to  place  a 
cipher  on  the  right  of  the  1  ?  Answer  similar  questions  in  regard 
to  the  first  figures  in  29,  36,  45,  19,  58,  67,  71. 

7.  How  many  thousands,  hundreds,  tens,  and  units  are  expressed 
in  each  of  the  following  numbers  :  3624,  5781,  9010,  8107  ? 


8.  Express  by  figures  the  number  of  sticks  represented  in  the 
1st  or  units'  group  ;  also  in 

the  3d  the  4th  and  3d  the  2d  and  1st  the  3d,  2d,  and  1st 

the  2d  the  3d  and  1st  the  4th  and  1st  the  4th,  3d,  and  1st 

the  4th  the  3d  and  2d  the  4th  and  2d  the  4th,  2d,  and  1st 

What  number  is  represented  in  all  the  groups  together  ? 


Note. — Pupils  should  prepare  suitable  objects  for  such  illustrations.  Bundles 
of  tens,  hundreds,  etc.,  with  single  objects,  should  often  be  arranged  promiscu- 
ously, and  the  learner  be  required  to  write  the  number  in  figures.  Let  him  observe 
that  we  may  estimate  the  value  of  the  bundle  by  its  size,  but  whether  a  digit  repre- 
sents tens  or  thousands  depends  on  the  place  it  occupies. 

9.  Write  in  columns,  of  ten  each,  all  the  numbers  from  one 
hundred  to  one  hundred  fifty-nine.  Also,  from  two  hundred 
fifty-one  to  three  hundred.  Also,  from  seven  hundred  eighty- 
three  to  eight  hundred  thirty-two.  Also,  from  six  hundred 
twenty-two  to  seven  hundred  one. 

10.  Write  all  the  numbers  from  one  thousand  six  hundred 
seventy-three  to  one  thousand  seven  hundred  two.  Write  in  col- 
umns of  ten  numbers  each. 


14  STANDARD  ARITHMETIC. 

10.  After  Thousands.— 1.  We  have  thus  learned,  First, 
the  names  of  the  orders  to  thousands ;  Second,  that  ten  of  any 
order  make  one  of  the  next  higher ;  and,  Third,  that  the  order 
of  a  figure — that  is,  whether  it  represents  units,  teus,  hundreds, 
or  thousands — is  known  by  the  place  which  it  occupies,  num- 
bered from  the  right. 

2.  When  we  reach  a  thousand  we  begin  to  count  the  thousands 
as  we  did  the  units  or  ones  :  that  is,  we  count  1  thousand,  2  thou- 
sand, up  to  999  thousand,  and  when  we  have  a  thousand  thousand 
we  call  the  number  one  million.  Millions  we  count  in  the  same 
way :  that  is,  1  million,  2  millions,  etc.,  up  to  999  millions. 
When  we  reach  a  thousand  millions,  we  call  the  number  a  billion. 
Billions  we  count  in  the  same  way,  so  trillions,  quadrillions,  etc. 

3.  In  writing  these  numbers  we  might  write  the  number  of 
thousands  as  we  do  the  numbers  up  to  a  thousand,  and  attach  the 
word  thousand  to  each  number ;  thus,  8  thousand,  76  thousand, 
999  thousand,  etc.,  etc.  But  just  as  we  avoid  writing  the  words 
units,  tens  and  hundreds,  by  giving  to  each  order  its  place,  so  we 
avoid  writing  the  word  thousand  by  giving  thousands  the  three 
places  to  the  left  of  hundreds.  In  the  same  manner  we  give 
millions  the  three  places  to  the  left  of  thousands. 

4.  In  this  way  it  comes  that,  when  more  than  three  figures  are 
employed  to  express  any  whole  number,  they  are  divided  into 
groups,  the  first  of  which,  numbering  from  the  right,  is  used  to 
denote  any  number  from  1  to  999  units  ;  the  second,  from  1  to  999 
thousand ;  the  third,  from  1  to  999  million,  etc.  These  groups 
are  called  Periods,  and,  for  convenience  in  reading,  are  sometimes 
separated  from  each  other  by  commas. 

5.  Thus,  beginning  at  the  right,  we  have  the  first ^period,  con- 
sisting of  ones,  tens  and  hundreds  of  units  ;  the  second  period, 
ones,  tens  and  hundreds  of  thousands  ;  the  third  period,  ones,  tens 
and  hundreds  of  millions.  The  fourth  period  is  that  of  billions  ; 
the  fifth,  trillions  ;  the  sixth,  quadrillions  ;  each  period  contain- 
ing ones,  tens,  and  hundreds  of  that  period. 


NOTATION  AND  NUMERATION  15 

li.   Numeration  Table. 


S3  °3S  2    a    »  So*  So*  So 

B     ~  3     S     r  3     S     n  3     £     a  Said  3     SI 


WHO    WHO  WHO  WHO  WHO  WHO 

Quadrillions  Trillions  Billions  Millions  Thousands  Ones 

Sixth               Fifth  Fourth  Third  Skcond  First 

741   852  963  074  197  581 

7   299  977  802  814  356 

865  243  576  006  050 

182   653  578  0  00  769  378 

31  840  999  000  642 

10   000  000  996  342  876 

875   763  954  123  508  627 


EXERCISES    IN     READING    AND    WRITING     NUMBERS. 

Read  the  foregoing  numbers,  consulting  the  headings,  till  you 
get  accustomed  to  the  names  of  the  periods. 

Read  also  as  follows,  or  as  may  be  directed  by  the  teacher. 

a.  Read  the  tens'  and  ones'  columns  in  the  first  period  on  the 
right. 

b.  Read  all  the  numbers  of  the  ones'  period. 

c.  Read  the  right-hand  column  in  the  thousands'  period. 

d.  Read  the  tens'  and  ones'  columns  in  the  thousands'  period. 

e.  Read  the  two  right-hand  periods. 

/.  Read  the  right-hand  column  of  the  millions'  period  with 
the  left  of  thousands,  as  hundred  thousands. 

g.  Read  the  millions'  and  ones' periods,  omitting  the  thousands' 
period  as  if  filled  with  ciphers. 

Note. — These  exercises  may  be  varied  to  almost  any  extent. 

1.  Read  these  numbers : 

10,000        83,000         60,000  75,000  150,000  756,000 

23,600        14,900        46,300  65,100  294,000  632,480 

47,225         83,720        85,493  62,340  392,500  290,405 

80,027        90,008        84,003  60,050  576,168  161,002 


790,000 

206,960 

410,000 

359,200 

941,000 

562,387 

684,210 

678,800 

143,576 

500,007 

246,890 

635,794 

16  STANDARD  ARITHMETIC, 

2=  Write  and  read : 
800,000  450,000 

743,000  200,000 

375,670  237,090 

135,791  987,654 

3.  Write  and  read  2,000,000  ;  5,000,000  ;  7,000,000  ;  4,000,000. 
Fill  the  places  occupied  by  ciphers  with  any  digits  yon  choose, 
and  then  read  the  numbers  thus  formed.    Do  this  in  various  waySo 

4.  Copy  the  following  numbers,  and  read  them  ;  then  erase  the 
digit  at  the  right  hand,  and  arrange  the  periods  anew,  by  placing 
the  commas  where  they  should  be,  and  read  : 

1,635,987  416,429,863  134,764,211  7,763,664  29,876,354 

Continue  this  exercise  by  erasing  the  digits  one  by  one,  and  j)oint- 
ing  off  the  periods  correctly. 

5.  How  many  tens,  hundreds,  thousands,  ten- thousands,  hun- 
dred-thousands and  millions  are  expressed  in  those  places  respec- 
tively, in  the  numbers  of  Exercise  4  ? 


Review   Exercises. 

Hundreds.— l.  Write  300  and  20  and  7  as  one  number,  ex- 
pressed by  three  figures.     In  the  same  way  write  : 

400  and  50  and  3  ;  100  and  70  and  6  ;  200  and  80  and  2 ; 
300  and  60  and  6;  600  and  90  and  4;  900  and  10  and  1. 

2.  Write  :  1  hundred  4  tens  6  ones;  2  hundreds  6  tens  3  ones; 

1        »«        8    "    7     "       4        "         7     "    5     " 
1        "        9    "     2     "       8        "  0     "    0     " 

3.  Read; 

527  723  168  365  134  524  340 

729  792  843  290  209  902  299 

313  901  910  109  953  646  728 

Thousands. — 4.  Make  the  following  numbers  larger  by  one 
thousand  :  328,  456,  508.    By  three  thousand.     By  five  thousand. 

5.  What  number  is  next  greater  than  1599,  3019,  4091,  8400, 
6379,  4599,  9999,  8765,  9109,  3099,  4098  ? 


NOTATION  AND   NUMERATION  17 

6.  What  number  comes  next  before  2000,  7000,  4600,  5060, 
3010,  2790,  8970,  1000,  1010,  7801  ? 

7.  Count  and  write  from  996  to  1006;  from  3189  to  3200; 
from  7990  to  8012  ;  from  3001  back  to  2989  ;  etc. 

8.  How  many  hundreds  and  tens  in  43  tens  ?  In  68,  37,  56, 
27,  49,  168,  434  tens  ?  How  many  in  386  units  ?  In  468,  125, 
632  units  ?    In  354,  538,  624  tens  ? 

9.  How  many  thousands  and  hundreds  are  in  25  hundreds  ? 
In  61,  52,  47,  56  hundreds  ?  How  many  in  250  tens  ?  In  310, 
161,  289,  364,  543  tens  ?    How  many  in  6987  units  ? 

Tens  and  Hundreds  of  Thousands. — 10.  Prefix  first  twenty, 
then  sixty,  then  forty  thousand  to  438,  132,  596,  100. 

11.  What  number  next  greater  than  25,999?  130,109? 
199,999?    888,889?     986,290?     18,400?     689,999? 

12.  What  number  next  less  than  300,001  ?  700,000  ?  147,000  ? 
354,989?    500,790?     100,000?    600,999?    489,123?    500,000? 


Definitions. 

12 .  A  unit  is  one  of  any  order  or  kind. 

13 .  A  number  is  a  unit  or  collection  of  units. 

14.  Notation  is  the  expression  of  number  by  figures  or  letters, 

15.  Numeration  is  the  reading  of  numbers  written  in  letters 
or  figures. 

16.  All  the  digits  have  a  Simple  Value  and  a  Local  Value. 
A  simple  value,  when  they  represent  units  or  ones  ;  a  local  value, 
when  used  to  express  tens,  hundreds,  etc.  This  value  is  called 
local  because  it  depends  on  the  place  which  the  digit  occupies  (its 
locality). 

17.  The  nine  digits  are  signs  of  number,  hence  they  are  called 
Significant  Figures.  In  this  sense,  the  cipher  "0"  is  not  a 
significant  figure. 


18  STANDARD  ARITHMETIC. 

Roman   Notation. 

18.  The  following  table  gives  a  complete  view  of  a  method 
of  representing  numbers  by  letters.  This  is  called  the  Roman 
method  because  first  used  by  the  Eoman  people. 

Table. 

Uuits. 


IV 

V 

VI 

VII 

VIM 

IX 


Note  1. — It  will  be  noticed  that  in  writing  four  and  nine  of  each  order,  a  letter 
of  less  value  is  placed  before  one  of  greater  value.  In  this  case  the  less  value  is 
deducted  from  the  greater.  Thus,  XL  (ten  less  than  fifty)  is  written  for  XXXX. 
CD  (one  hundred  less  than  five  hundred)  is  written  for  CCCC,  etc.  This  mode  of 
abbreviation  is  common,  not  universal. 

Note  2. — A  bar  over  a  letter,  or  combination  of  letters,  increases  its  value  a 
thousand  times. 

For  writing  numbers  in  Roman  numerals,  we  have  the  following 

19,  Mule. — Write  the  several  terms  in  order  as  given  in  the 
table. 

EXERCISES    IN    THE     ROMAN     NOTATION. 

1.  Eead  XXIV,  IX,  XIX,  XV,  XIV,  LX,  XLIV,  LXXXIX, 
XO,  XCIX,  CCI,  CCCXOIX,  CD,  CDLVIII,  CDLIX. 

2.  DV,  CDXCIX,  DXLVI,  DCCCIX,  CMXCIX,  MD,  MIX, 
LIX,  DIX,  MDCCCLXXXIV,  MCDLXX. 


Thousands. 

Hundreds. 

Tens 

M 

c 

X 

MM 

cc 

XX 

MMM 

ccc 

XXX 

IV 

CCCC 

or  CD 

XL 

V 

D 

L 

VI 

DC 

LX 

VII 

DCC 

LXX 

VIII 

DCCC 

LXXX 

IX 

CM 

xc 

X- 

-10000 

3.  Write  in  Roman  numerals,  54,  72,  83,  59,  119,  72,  38,  49, 
63,  98,  75,  69,  43,  91,  108,  319,  444,  333,  991,  3847,  2563,  3482. 


CHAPTER    II. 

ADDITION. 

Examples. — l.  A  hunter  shot  6  rabbits  on  Monday,  7  on  Tues- 
day, 8  on  Wednesday,  but  only  one  on  Thursday.  How  many 
rabbits  did  he  shoot  ? 

2.  Charles  is  9  years  old.  How  old  will  he  be  in  6  years  ?  In 
5  years  ?  In  8  years  ? 

3.  Fred  had  8  dollars  in  his  bank ;  he  received  7  more  on  his 
birthday,  and  4  at  Christmas.     How  much  had  he  then  ? 

4.  Grandfather  was  53  years  old  when  his  grandchild  was  born. 
How  old  is  he  now  that  his  grandchild  is  9  years  old  ? 

5.  The  sun  rose  at  6  o'clock  this  morning ;  that  was  3  hours 
ago.  What  o'clock  is  it  now  ?  What  o'clock  5  hours  after  sun- 
rise ?  4  hours  ?     6  hours  ? . 

6.  William  read  7  pages  in  the  morning,  3  in  the  afternoon, 
and  4  in  the  evening.     How  many  pages  did  he  read  that  day  ? 

7.  Sarah  goes  up  and  down  stairs  8  times  in  the  morning,  and 
5  times  in  the  afternoon.     How  many  times  in  the  day  ? 


8.  Count  to  one  hundred. — Count  by  twos  to  100. — Count  by 
threes  to  99.  Count  by  fours  to  100. — Count  by  fives  to  100. — 
Count  by  sixes  to  96. — Count  by  sevens  to  98. — Count  by  eights 
to  96. — Count  by  nines  to  99. 

The  pupil  may  first  write  the  result  of  each  successive  addition,  and  afterward 
go  through  the  exercise  orally. 

9.  How  many  units  in  10  twos  ?  (Count  by  2's  till  you  find 
out.)     How  many  in  10  threes  ?    In  10  fours?    In  10  fives  ?  etc. 


20  STANDARD  ARITHMETIC. 

10.  Draw  lines  upon  your  slate,  so  as  to  divide  it  like  a 
checker-board,  but  make  ten  squares  instead  of  eight  in  each 
row,  and  as  you  count  by  l's,  write  the  results  in  the  first  line 
of  squares  from  left  to  right ;  as  you  count  by  2's,  write  the  re- 
sults in  the  second  line  of  squares  ;  as  you  count  by  3's,  write  the 
results  in  the  third  line,  and  so  on. 


Definitions. 

20.  Addition  in  arithmetic  is  a  process  of  finding  the  sum 
of  two  or  more  numbers. 

21.  Signs.— 1.  The  sign  +  is  read  plus,  and  indicates  addi- 
tion ;  thus,  5  +  3  means  5  and  3  more. 

2.  The  sign  =  is  read  equals,  or  is  equal  to ;  thus,  5  +  3  =  8 
is  read,  5  plus  3  equals  or  is  equal  to  8. 


ORAL     EXERCISES. 

Write  on  slate  or  paper  these  two  lines  of  figures. 

4,  7,  2,  8,  8,  5,  6,  3,  9,  2,  6,  6,  8,  7,  9,  5,  3,  3,  4,  2,  5, 

5,  3,  7,  7,  5,  5,  4,  6,  9,  9,  8,  4,  4,  2,  3,  8,  5,  4,  9,  6,  7. 

11.  To  each  number  represented  add  2,  add  4,  add  6,  add  8. 

Caution. — Do  not  say  4  and  2  are  6,  but  speak  only  the  results,  as  6,  9,  etc. 
In  13  (below)  give  results  directly,  as  11,  9,  10,  16,  etc. 

12.  Add  3,  add  5,  add  7,  add  9,  to  each  one. 

13.  Add  the  first  to  the  second  ;  add  the  second  to  the  third, 
etc.,  beginning  at  the  left — beginning  at  the  right. 

14.  Add  each  number  in  the  lower  line  to  the  one  above  it, 
proceeding  first  from  left  to  right,  and  then  from  right  to  left. 


15.  4  +  5  +  6=  20.  2  +  3  +  4  +  5=  25.  4  +  4  +  4  +  4= 

16.  5  +  6  +  7=  21.  3  +  1+4  +  8=  26.  6  +  3  +  6  +  3  = 

17.  6  +  2  +  8=  22.  4  +  7  +  6  +  3=  27.  5  +  5  +  5  +  5  = 

18.  4  +  9-l-3=  23.  9  +  3  +  2  +  3=  28.  7  +  2  +  7+2= 

19.  7  +  4  +  9=  24.  7  +  4  +  6  +  3=  29.  8  +  4  +  5  +  4= 


ADDITION. 


21 


30.  6  +  5  +  3: 

8  +  4  +  2= 

9  +  7  +  4: 
5  +  8  +  3: 

7  +  4  +  7: 

33. 

10  +  6: 
30  +  5: 
40  +  4: 
50  +  3  = 
60  +  2: 
80  +  7: 


31.  5  +  4  +  8  +  2  = 

2+4+6+8= 
3+5+7+4= 
3+6+3+7= 
8  +  l  +  J*  +  2  = 


34. 

20  +  9= 
50  +  7= 
70  +  5= 
90  +  3= 
30  +  2  = 
60  +  8= 


35. 

90  +  2: 
80  +  4: 
70  +  6: 
60  +  3: 
50  +  5: 
40  +  9: 


32.  5  +  3  +  4  +  2: 

9  +  7+1+2: 

6  +  4+4+3: 
9  +  4+4+3= 
4+9.+  3  +  4= 

36. 

20  +  2+4= 
40  +  3  +  2  = 
60  +  4  +  5= 
80  +  5  +  2= 
30  +  1  +  7= 
70  +  2  +  6= 


37-117.  Add  1,  2,  3,  etc.,  up  to  9,  separately  to  each  number 
in  each  line.     Observe  the  units  of  the  results. 


1  11  21  31  41  51  61  71  81 

2  12  22  32  42  52  62  72  82 


This  may  be  done  orally, 
or  on  the  slate,  thus : 


13  23  33  43  53  63  73  83  (37.)  1  +  1  = 


14  24  34  44  54  64  74  84 

15  25  35  45  55  65  75  85 

16  26  36  46  56  66  76  86 

17  27  37  47  57  -67  77  87 

8  18  28  38  48  58  68  78  88 

9  19  29  39  49  59  69  79  89 


11  +  1  = 
21+1  = 
31  +  1  = 
41+1  = 
51  +  1  = 
etc. 


118. 

32  +  9= 
43  +  8= 
54  +  7= 
65  +  6  = 
76  +  5= 
87+4= 
91  +  8= 


119. 

33  +  9= 

44  +  8= 
55  +  7= 
66  +  6= 
77+5  = 
88  +  4= 
22  +  7= 


120. 

87  +  6  = 
78  +  5= 
45  +  8= 
54+7= 
65  +  9= 
56+4= 
37  +  8= 


(117.)  9  +  9= 
19  +  9= 
29  +  9  = 
39  +  9= 
49  +  9= 
59  +  9= 
etc. 

121. 

56  +  8= 
65  +  6= 
29  +  9= 

92  +  5  = 
87  +  7= 
78  +  6= 
47  +  6= 


122-127.  Add  6  to  21,  18,  36,  48,  54,  63,  17,  82,  88.  Add 
also  8  ;  4  ;  5  ;  7  ;  9. 

123-131.  Increase  the  numbers  14,  19,  23,  25,  48,  84,  56,  37, 
64,  83,  52,  38,  90,  87,  75,  61,  47,  79,  39,  59,  27,  69,  89,  by  4 ;  by 
6  ;  by  8  ;  by  10. 


22  STANDARD  ARITHMETIC. 

133-135.  Increase  each  of  the  numbers  23,  35,  48,  64,  56,  37, 
41,  90,  82,  52,  61,  73,  84,  59,  47,  36,  by  3  ;  by  5 ;  by  7 ;  by  9. 

Direction. — The  foregoing  exercises  should  be  so  thoroughly  practiced,  both 
orally  and  in  writing,  that  the  pupil  can  announce  the  sum  of  any  two  numbers 
expressed  by  single  digits  as  readily  as  he  can  read  them. — If  he  has  to  count  his 
fingers  in  addition,  he  can  proceed  but  slowly.  He  might  as  well  spell  every  word 
as  he  reads,  or  crawl  on  his  hands  and  knees  instead  of  walking.  Again,  if  he  says 
"9  and  7 "are  16,"  he  uses  five  words  where  one,  "sixteen,"  would  be  better. 

Applications. — 136.  An  hour  has  60  minutes,  and  a  half-hour 
has  30.     How  many  minutes  are  there  in  one  hour  and  a  half  ? 

137.  There  were  hanging  on  a  Christmas-tree  10  oranges,  20 
apples,  30  nuts,  20  sugar-plums.     How  many  gifts  in  all  ? 

138.  There  are  at  work  in  a  factory  40  men  on  the  ground- 
floor,  30  on  the  second  floor,  20  on  the  third  floor,  and  7  in  the 
office.     How  many  men  are  at  work  in  the  factory  ? 

139.  Grandmamma  is  60  years  old,  mamma  30,  and  I  am  7 
years  old.     What  is  the  sum  of  our  ages  ? 

140.  An  overcoat  costs  30  dollars,  a  coat  20  dollars,  a  vest  4, 
and  a  pair  of  trousers  8  dollars.  How  much  does  the  whole  suit 
cost? 

141.  A  fisherman  caught  in  his  net  36  pike,  30  bass,  and  10 
trout.     Can  you  tell  how  many  fish  he  caught  ? 

142.  A  butcher  bought  two  calves ;  one  weighed  53  pounds, 
the  other  47.     How  much  did  they  weigh  together  ? 


ORAL     EXERCISES. 

Direction. — In  adding,  do  not  say  (see  1st  example)  4  and  5  are  9  and  6  are  15, 
etc.,  but  give  results  at  once ;  thus,  4,  9,  15,  23,  etc. 

143-168.  Add  by  columns  and  lines. 

4+5+6+8+9+4+3+7=  9  +  6  +  8  +  2+7  +  5  +  9  +  3  = 

3+2+1+9+7+6+2+8=  6+7+4+9+3+1+7+5= 

8+4+3+9+5+7+8+6=  8+5+3+6+2+8+4+7=" 

6  +  6  +  8  +  3  +  7+4  +  4  +  5=  34.5  +  4  +  9  +  7  +  2  +  8  +  2= 

7+8+2+7+6+9+6+5=  3+4+2+5+3+1+6+2= 


ADDITION. 

169. 

170. 

171. 

172. 

10  +  40= 

90  +  10= 

20  +  70= 

40  +  15: 

20  +  50= 

80+10= 

30  +  60= 

30  +  26= 

30+40= 

60  +  30= 

40  +  50= 

60  +  38; 

40  +  50  = 

70  +  20= 

50  +  30= 

50  +  47 

50  +  30= 

50+40= 

60  +  15  = 

70  +  28: 

60+40= 

40+40= 

70  +  26  = 

40+48 

70  +  20= 

80  +  20= 

80  +  17= 

60  +  37 

90  +  30= 

20  +  60= 

20+45= 

70+46: 

50  +  50= 

30  +  30= 

40  +  4:1  = 

80  +  11 

23 


174.  30  +  20  +  10  +  30= 

175.  20  +  10  +  30  +  20= 

176.  10  +  30  +  20  +  30= 

177.  40  +  20  +  30  +  10= 


173. 

56  +  30= 
44  +  20= 
38  +  60: 

29  +  50: 
14+80= 
63  +  30: 

74  +  20: 

88+40= 

30  +  53: 

178.  30+15  +  20  +  15  +  10  +  3+  7= 

179.  10  +  25  +  30  +  10  +  20  +  2+  2  = 

180.  20  +  25  +  10  +  15  +  10  +  3  +  14= 

181.  15  +  25  +  30  +  10  +  12  +  4+  4= 


Suggestions. — If,  from  this  point  to  the  rule  on  page  28,  the  examples  seem 
too  difficult,  they  may  be  omitted,  to  be  taken  up  under  the  rule,  but  let  the  oral 
work  be  carried  as  far  as  possible.  The  learner  who  is  left  to  himself  to  work  out 
all  his  exercises  on  the  slate  is  apt  to  form  habits  fatal  to  accuracy. 


Applications. — 182.  There  are  25  girls  and  23  boys  in  a  school- 
room.    How  many  pupils  in  all  ? 

183.  On  one  side  of  Blair  Street  there  are  34  houses,  on  the 
other  side,  59.     How  many  on  both  sides  ? 

184.  Mr.  H.  bought  a  horse  for  73  dollars ;  he  sold  it  and 
gained  19  dollars.     For  how  much  did  he  sell  it  ? 

185.  Our  house  has  17  windows,  that  of  one  neighbor  has  26, 
and  that  of  another  neighbor  has  13.  How  many  windows  are 
there  in  the  three  houses  ? 

Exercises. — 186-206.  Add  rapidly,  by  columns  and  lines,  giving 
results  only  : 

4+6+4+8+7+4+9+4+3+5+2+3+1+8+9= 
8+5+3+9+6+8+5+3+6+7+9+7+6+9+1= 
4+9+5+6+9+7+6+2+1+8+3+4+2+3+9= 
7+6+5+4+3+2+1+9+7+5+6+4+9+4+8= 
4+3+2+1+2+3+4+5+6+7+8+9+7+6+7= 
8+4+6+9+1+7+3+5+7+9+6+8+7+5+6= 


24 


STANDARD  ARITHMETIC. 


Note. — Let  the  pupil  illustrate  examples  207  to 
226  by  the  use  of  buttons,  acorns,  or  other  objects 
which  he  can  tie  into  bundles  or  string  together  in 
collections  of  ten;  or  let  him  make  marks  upon 
the  slate  such  as  these  at  the  right,  designed  to 
illustrate  example  207. 

Thus,  understanding  well  the  nature  of  the 
thing  to  be  done,  he  will  need  no  rule  for  the  sim- 
ple operations  here  required.  Let  him  first  add 
the  tens,  and  to  the  sum  let  him  add  the  numbers 
in  units'  place. 

207.  17  +  15+10  +  18  +  19=  217. 

208.  14  +  16  +  18  +  20  +  13=  218. 

209.  12  +  13  +  20  +  14+16=  219. 

210.  20  +  19  +  11  +  18  +  13=  220. 

211.  18  +  15  +  14  +  17  +  16=  221. 

212.  12  +  14  +  16  +  13+15=  222. 

213.  15  +  17  +  19  +  16  +  14=  223. 

214.  19  +  12  +  13  +  17  +  16=  224. 

215.  17  +  18  +  14  +  17  +  18=  225. 

216.  20  +  19  +  19  +  12  +  13=  226. 


WWW//  /r 
MMM  /s 
MM  /o 
MM  M//  /s 
MMMM/& 


12  +  15  + 

18  +  10  + 

19  +  17  + 

13  +  20  + 
23  +  17  + 

8  +  12  + 

11  +  17  + 
27  +  19  + 

12  +  24  + 

20  +  13  + 


9  +  10  +  13  = 
17+  9  +  18= 

15  +  13  +  11  = 

7  +  16  +  12  = 
18+  2  +  15  = 
13  +  19  +  21  = 
28+  6  +  18= 

8  +  20  +  16: 

18+     6  +  17: 

16  +  10  +  25: 


Suggestion. — Exercises  in  numeration  should  precede  the  following  examples. 

Applications. — 227.  There  were  at  a  party  50  gentlemen,  60 
ladies,  and  70  children.     How  many  people  were  there  ? 

228.  A  farmer  raised  80  bushels  of  wheat;  39  bushels  of  oats, 
and  10  bushels  of  barley.     How  many  bushels  in  all  ? 

229.  There  are  in  an  orchard  63  plum-trees,  75  apple-trees, 
and  11  peach-trees.     How  many  trees  in  all  ? 

230.  A  book-case  has  on  the  first  shelf  48  books,  on  the  second 
57,  and  on  the  third  75.     How  many  on  the  3  shelves  ? 

231.  There  are  68  boys  in  one  room  of  a  school-house,  73  girls 
in  another,  and  87  girls  and  boys  in  a  third.  How  many  pupils 
are  there  in  the  school  ? 

232.  The  first  book  of  Moses  has  50  chapters,  the  second  40,  the 
third  27,  the  fourth  36,  and  the  fifth  34.  How  many  in  the  5  books  ? 


ADDITION. 


25 


Add  without  the  use  of  the  slate : 
233.  50  +  60=     234.  40  +  60  +  30=     235. 


60  +  80= 
70  +  40= 
80  +  30= 
90  +  50= 
40  +  90= 


50  +  30  +  50= 
60  +  90  +  70= 
70  +  40  +  90= 
80  +  70  +  40= 
90  +  50  +  60= 


80  +  38= 
70  +  59= 
90  +  67= 
50  +  74= 
60  +  83  = 
70  +  92= 


236. 


64  +  50= 
76  +  60= 
85  +  90= 
59  +  70= 
68  +  80= 
75  +  40= 


Note. — The  examples  in  235  and  236  require  only  one  oral  step,  that  is,  the 
direct  announcement  of  the  result ;  as,  for  instance,  in  adding  80  and  38,  think 
80  and  30  (=110)  and  8,  but  say  at  once  118.  In  examples  237  to  241,  two  steps 
are  enough ;  thus,  in  adding  59  and  32,  first  think  59  and  30,  and  say  89,  then 
89  and  2,  and  say  91.  In  examples  242  to  244,  four  steps  may  be  necessary  for 
the  learner ;  thus,  in  adding  25,  38,  and  49,  say  55,  63,  103,  112. 


237.         238. 

239. 

240. 

241. 

59  +  32=      36  +  42= 

25  +  22  = 

57  +  28= 

26  +  34= 

44  +  27=  '    47  +  33= 

24  +  35= 

28  +  39= 

24  +  16= 

36  +  24=      38  +  24= 

33  +  19= 

45  +  37= 

74  +  24= 

47  +  25=      58  +  25  = 

42  +  35  = 

36  +  38= 

25  +  33= 

88  +  26=      64  +  26= 

53  +  43= 

25  +  47= 

47  +  29= 

45  +  37=      47  +  27= 

65  +  26= 

34  +  39= 

58  +  33= 

62  +  25=      29  +  34= 

35  +  36= 

27  +  48= 

244. 

34  +  28= 

242. 

243. 

65  +  36  +  3  = 

,  85  +  47  +  5  = 

25  +  38  +  4S 

1= 

85  +  74  +  5= 

83  +  68  +  6= 

97  +  23  +  62 

67  +  63  +  7= 

94  +  42  +  9= 

56  +  44  +  68 

58  +  54  +  6= 

86  +  29  +  3= 

83  +  28  +  25 

73  +  72  +  9= 

92  +  56+5  = 

62  +  37  +  27 

68  +  45  +  4= 

75  +  65  +  6  = 

56  +  48  +  39 

i  — 

245-259.  Add  by  columns  and  lines  : 

40  +  60  +  30  +  70  +  80  +  90  +  60  +  80  +  80  +  30 
50  +  80  +  90  +  80  +  40  +  50  +  90  +  70  +  20  +  50 
60  +  40  +  70  +  90  +  20  +  70  +  70  +  60  +  70  +  30 
70  +  90  +  20  +  60  +  30  +  30  +  40  +  50  +  90  +  70 
80  +  70  +  60  +  50  +  40  +  30  +  20  +  40  +  30  +  50 


STANDARD  ARITHMETIC. 


Suggestions  for  Blackboard  Exercises. 

22.  In  drill  exercises,  the 
double  star  affords  some  ad- 
vantages over  the  circle,  and 
at  the  same  time  facilitates  the 
learning  of  the  several  series 
arising  from  successive  addi- 
tions of  %%  3's,  4's,  etc. 


Direction.  —  Beginning  at 
the  unit  figure  of  any  given 
number,  the  unit  figures  of 
the  successive  sums  will  be 
found  as  follows  : 

1.  In  adding  3's,  at  the 
next  point  to  the  right,  and  so  on  ;  in  adding  Ts,  at  the  next  to 
the  left. 

2.  In  adding  6's,  at  the  second  point  to  the  right,  and  thus  on, 
from  point  to  point,  of  the  same  star.  In  adding  4's,  at  the  second 
point  to  the  left,  and  so  on. 

3.  In  adding  9?s,  at  the  third  point  to  the  right. 

4.  In  adding  2's,  at  the  fourth  point  to  the  right,  and  thus  on 
(following  the  line  at  the  right  of  the  last  unit  figure).  In  adding  8's,  at  the 
fourth  point  to  the  left  (following  the  line  at  the  left  of  the  last  unit  figure). 

5.  In  adding  5's,  at  the  point  directly  opposite  the  unit  figure 
of  the  given  number,  and  thus  to  and  fro. 

23.  Other  Uses  of  the  Figure. — A  suitable  number  being  writ- 
ten at  the  center,  the  numbers  at  the  points  can  be  combined  with 
it,  in  addition,  subtraction,  multiplication,  or  division,  as  may  be 
desired.  The  number  at  the  center  being  changed  from  time  to 
time,  there  is  no  end  to  the  variety  of  exercises  that  may  thus  be 
had  at  little  expense  of  time  or  labor  on  the  part  of  the  teacher. 
Exercises  in  common  and  decimal  fractions  may  be  given  in  the 
same  way. 


ADDITION. 


27 


Addition  of  Higher  Orders. 

24"«  Note. — The  illustrations  of  this  book  are  not  intended  to  be  merely  ob- 
served and  read  about,  but  they  are  designed  to  picture  to  the  eye,  as  far  as  possible, 
the  actual  work  which  it  is  intended  shall  be  done  by  the  pupils  with  objects.  These 
objects  should  be  supplied  by  the  school  authorities,  or,  with  slight  suggestions  by 
the  teacher  as  to  what  is  best  or  most  available,  according  to  the  circumstances  of 
the  school,  they  may  be  brought  in  by  the  pupils.  They  should  be  as  large  as  pos- 
sible, and  yet  not  inconvenient  to  handle  in  great  numbers. 


SLATE     WORK. 


Example.— Find  the  sum  of  738,  236  and  573. 

One  who  knows  nothing  more  of  arithmetic  than  how  to  count 
to  ten  might  find  the  sum  of  these  numbers  by  some  such  means 
as  the  following  : 

Suppose  that  he  has  a  large  num- 
ber of  sticks,  some  of  them  tied  up 
in  bundles  of  ten,  and  some  in  bun- 
dles of  a  hundred  each,  and  that  he 
has,  besides,  some  single  sticks.     If 
these  were  placed  in  rows  or  shelves, 
as  in  the  picture  at  the  right,   he 
might  count  first  the  single  sticks, 
taking  them  in  his  hand  as  he  does 
so,  and  when  he  has  reached   ten, 
tie  them  in  a  small  bundle,  leaving 
the  remaining  single   sticks  at    the 
right  on  the  shelf  below.     He  could 
then  count  the  bundle  which  he  had 
just  made,  with  the  tens' 
bundles  on  the  shelves, 
and  tie  each  ten  of  these 
bundles  into  larger  bun- 
dles of  a  hundred  each, 
and  leaving  the  odd  bun- 
dles of  tens  on  the  shelf  below  where  they  formerly  were,  he  could 


ii 


B  - 


STANDARD  ARITHMETIC. 


count  all  the  bundles  of  hundreds  together, 
ten  of  these  larger  bun- 
dles into  one,  he  would 
have  the  sticks  arranged 
as  here  represented ; 
that  is,  one  bundle  con- 
taining a  thousand,  five 
of  a  hundred  each,  four 
of  ten  each,  and  seven 
single  sticks. 

25.    The  foregoing 
method  is  the  same  as  that  which  is  indicated  in  the  following 
arithmetical  process : 

<&u?ztd  (7Atm&  ofcnJ.  <2&U&.     ct/itntJ.  (?/&*zd.  C&**4.  fl&Utt 


r 


3 


*u 


7Z 


3 


els  /      //+/      j+v      7 


Having  seen  how  it  is,  that  this  process  really  produces  a 
number  equal  to  the  sum  of  the  numbers  added,  the  pupil  is 
prepared  for  the  rule  for  addition. 

26.  Bule.—l.  Arrange  the  numbers  to  be  added  so  that  the 
figures  of  the  same  order  shall  stand  in  the  same  column,  units 
under  units,  tens  under  tens,  and  so  on. 

2.  Begin  at  the  lowest  order,  and  add  each  column  separately. 
If  the  sum  of  any  column  is  less  than  10,  write  it  underneath.  If 
it  is  equal  to  or  greater  than  10,  place  the  right-hand  figure  of 
the  sum  under  the  column  added,  and  unite  the  left-hand  term  or 
terms  with  the  next  column. 

Vroof. — In  order  to  be  quite  sure  that  the  addition  is  correct, 
add  each  column  both  upward  and  downward.  If  the  two  results 
are  the  same,  there  is  little  danger  of  error. 


ADDITION. 


29 


SLATE     EXERCISES, 

Examples  l-ll.  Find  the  sum  of 


27 

37 

29 

68 

68- 

92 

53 

27 

57 

91 

84 

58 

28 

36 

43 

79 

72 

67 

72 

46 

24 

37 

90 

64 

39 

29 

25 

68 

85 

28 

90 

56- 

62 

56 

47 

76 

46 

97 

73 

39 

57 

51 

45 

78 

12-22.  Find  the 

sum  of 

48 

83 

69 

21 

43 

96 

81 

49 

36 

85 

20 

47 

59 

56 

53 

28 

45 

47 

57 

17 

24 

38 

56 

62 

67 

38 

57 

51 

63 

42 

58 

66 

47 

29 

48 

58 

73 

46 

23 

86 

56 

32 

77 

93 

23 

-77.  Add  345  to  each, 

639 

542 

894 

457 

837 

910 

735 

628 

246 

802 

135 

429 

900 

312 

163 

524 

254 

736 

547 

749 

630 

757 

372 

713 

457 

298 

337 

698 

507 

192 

293 

394 

495 

345 

298 

110 

206 

387 

471 

289 

509 

135 

398 

211 

213 

361 

425 

357 

862 

135 

779 

387 

600 

731 

877 

78-88.  Add  together  the  numbers  in  each  column. 
89-93.  Arrange  each  line  of  numbers  in  column  and  add. 


94. 

95. 

96. 

97. 

93. 

99. 

100. 

Apples. 

Nuts. 

Oranges. 

Peaches. 

Lemons. 

Plums. 

Books 

136 

268 

204 

268 

301 

718 

593 

241 

194 

237 

473 

275 

629 

868 

217 

187 

168 

118 

478 

446 

687 

153 

253 

352 

323 

262 

537 

774 

302 

145 

249 

248 

164 

855 

956 

Add  the  following  numbers,  first  arranging  them  in  columns 

101.  125,  126,  138,  139,  140.  103.  83,  194,  56,  168,  473. 

102.  87,  9,  55,  394,  225,  194.  104.  336,  195,  987,  9,  11. 
105-114.  Add  by  columns.    Also  by  lines, 

2123  +  2364  +  7025+  428+     20  +  2103=. 

6354  +  2559+   843  +  1125+   359+     23= 

698  +  1994  +  1427  +  2496  +  2478+  437= 

1927+     49  +  7917  +  6579  +  1000  +  3706= 


30 


STANDARD  ARITHMETIC. 


115-124.  3254  +  4015  +  7348  +  1570+  439  +  7986= 

968+   916  +  3407  +  4630  +  1690+  375= 

725  +  1207+   197  +  1820+  420+       9= 

4302+  885  +  8329+       7  +  7756  +  8975= 

The  following  may  be  solved  first  without  the  use  of  the  slate. 
Only  results  should  be  pronounced.  (In  the  last  line,  Ex.  127, 
for  instance,  say  892,  952,  956.) 

125.    300  +  600=  126.  600  +  70  +  38= 

400  +  900=  700  +  50  +  49= 

500  +  100=  800  +  80  +  76= 

600  +  400=  900  +  90  +  47= 

700  +  700=  300  +  60  +  83= 

800  +  200=  200  +  80  +  72  = 


127.  300  +  56  +  84= 

400  +  72  +  65= 
500  +  83  +  49= 
600  +  64  +  57= 
700  +  53  +  88= 
800  +  92  +  64  = 


128-148.  A  dd  by  columns  and  by  lines. 
568  +  487+2000  +  5872  = 
435  +  675  +  6060  +  9321  = 
357  +  894  +  7009  +  7234= 
479  +  383  +  5800  +  5321  = 
692  +  596  +  3750  +  6947= 
824  +  776  +  4680  +  3579  *= 
546  +  484  +  8541+2468= 


149.  345,271  150.  4,391,002  151. 


4769  +  634  +  2465  = 
1250+  4  +  1975= 
3456+  27+  888= 
5861+  5+  935= 
9642+  95+  576= 
8347+  7  +  6491  = 
4936  +  1S3+   587= 


Find  the  i 

3i//77  Of 

.  345,271 

150.  4,391,002 

65,382 

686,975 

7,491 

68 

83,257 

4,937 

496,350 

53,286 

1,849 

487,659 

65,472 

7,321,445 

4,622,715 
9,874,963 
8,472,465 
8,010,706 
4,506,080 
432,741 
98,653 


152.  5,324,681 

1,964,735 

28,497 

68 

5,834 

4,793 

56,689 


153.  Add  768,  5,643,  12,354,  678,901,  5,847,  2,146,353,  975,321, 
64,387,510. 

154.  Add  nineteen,  ninety,  seventy  thousand  four  hundred 
eight,  87,  1,625,847,  269,751,  3,894,  twelve  hundred  sixty-one, 
5,050,050,  six  hundred  thousand  six. 


ADDITION.  31 

155.  Add  198,725,  918,273,  1,928,370,  4,354,651,  34,234,534, 
6,712,893,  647,  19,  1,345,  67,351. 

156.  Add  283,857,  two  thousand  twenty,  998,722,  five  mill- 
ions fifty  thousand  fifty,  eight  hundred  thousand  seven  hundred 
twelve,  27,  192,875,  909,090,  six  hundred  eight  thousand  four 
hundred  ten,  34,827,  fourteen  hundred  fourteen. 


157.  1,783 

158.  4,328 

159.  42,235 

160.  9,999,999 

19,456 

369,300 

10,305,236 

8,000,000 

5,788 

53,528 

84,165,352 

4,730,876 

94,374 

51,279 

1,236,536 

12,359,776 

100,855 

13,975 

163,021 

76,305,864 

456,788 

34,975 

29,363,987 

32,467,209 

872,543 

124,900 

37,903,210 

57,264,902 

321,354 

1,243,651 

16,988,710 

98,537,873 

Direction. — In  adding  these  columns,  do  not  say  (see  Ex.  157)  4  and  3  are 
*7,  and  8  are  15,  and  5  are  20,  and  4  are  24, etc.,  but  simply  speak  results  ;  thus: 
4,  7,  15,  20,  24,  32,  38,  41.  The  repetition  of  the  numbers  to  be  added  in- 
creases liability  to  error. 

Some  can  learn  to  add  mentally  numbers  of  even  three  places.  Treating  tbe 
tens  and  hundreds  as  units  and  tens  (see  note,  p.  25),  they  would  say,  in  Ex.  161, 
27,  57,  65  tens  =  650,  655,  661,  and  set  down  the  answer  at  once. 

27.  Adding  two  or  more  columns  of  figures  at  once  is  valu- 
able practice  in  "mental  arithmetic."     It  should  be  carried  as 
far  as  time  and  the  ability  of  the  pupil  will  permit. 
161-169.  275        369       .876        629        483        987        797        519        357 
386        625        529        875        759        654        686         982        678 


Examples  for  Practice  and  Review. 
Applications. — l.  I  gave  83  marbles  to  Lewis,  34  to  William, 
and  97  to  Charles.     How  many  did  I  give  away  ? 

2.  In  one  book  there  are  89  pages,  in  another  246,  and  in  a 
third  387.     How  many  pages  in  all  ? 

3.  A  certain  tract  of  land  was  divided  into  four  farms,  one 
containing  113  acres,  another  237,  a  third  180,  and  the  fourth 
320  acres.     How  many  acres  did  the  original  tract  contain  ? 


32  STANDARD  ARITHMETIC. 

4.  There  was  a  large  number  of  cents  in  a  bag.  I  took  out  of 
it  first  289  cents,  then  397,  then  478,  then  693,  and  then  I  found 
134  cents  left  in  the  bag.    How  many  cents  did  it  contain  at  first  ? 

5.  Our  school-house  contains  6  rooms.  In  room  No.  1  there 
are  25  boys  and  31  girls  ;  in  No.  2  there  are  18  boys  and  29  girls ; 
in  No.  3,  21  boys  and  37  girls  ;  m  No.  4,  29  boys  and  19  girls ; 
in  No.  5,  45  boys ;  in  No.  6,  53  girls.  How  many  boys  in  our 
school  ?    How  many  girls  ?    How  many  children  in  all  ? 

6.  In  a  certain  township  there  are  six  farmers.  The  first  has 
5  horses,  12  cows,  35  sheep,  and  20  hogs.  The  2d  has  5  horses, 
10  cows,  18  sheep,  and  12  hogs.  The  3d  has  3  horses,  6  cows,  27 
sheep,  and  9  hogs.  The  4th  has  4  horses,  8  cows,  25  sheep,  and 
14  hogs.  The  5th  has  1  horse,  2  cows,  and  6  hogs.  The  6th  has 
8  horses,  17  cows,  45  sheep,  and  27  hogs.  (1)  How  many  horses 
have  the  6  farmers  ?  (2)  How  many  cows  ?  (3)  How  many 
sheep  ?  (4)  How  many  hogs  ?  (5)  How  many  head  of  live 
stock  has  the  first,  the  second,  the  third,  the  fourth,  the  fifth, 
the  sixth  ?    (6)  How  many  head  of  live  stock  on  the  6  farms  ? 

For  the  solution  of  this  and  similar  examples,  follow  the  arrangement 
given  here.     It  is  called  "  Tabulating,"  or  arranging  in  tables. 


FARMERS 

HORSES 

COWS 

SHEEP 

HOGS 

NO.  HEAD 

The  First 

5 

12 

35 

« 

20 

The  Second 

5 

10 

18 

12 

The  Third 

3 

6 

27 

9 

The  Fourth 

4 

8 

25 

14 

The  Fifth 

I 

2 

— 

6 

The  Sixth 

8 

17 

45 

27 

Total 

Note. — If  the  sum  of  the  footings  in  the  last  line  is  not  equal  to  the  sum  of  the 
extensions  in  the  last  column,  the  work  is  incorrect.     Why  ? 


ADDITION.  33 

7.  Three  farmers  have  fruit-trees  as  follows  :  The  first,  72 
apple-trees,  108  peach-trees,  18  quince-trees,  16  plum-trees,  19 
cherry-trees.  The  second,  38  apple-trees,  219  peach-trees,  9 
quince-trees,  27  pear-trees,  38  plum-trees,  3  cherry-trees.  The 
third,  19  apple-trees,  and  43  peach-trees.  (1)  How  many  trees 
of  each  kind  ?  (2)  How  many  fruit-trees  has  each  farmer  ?  (3) 
How  many  trees  have  all  ? 

8.  In  the  year  1880,  Cincinnati  had  255,139  inhabitants ; 
Cleveland  160,146;  Toledo  50,137;  Columbus  51,647;  Dayton 
38,678;  Sandusky  15,838  ;  Springfield  20,730;  Hamilton  12,122  ; 
Portsmouth  11,321.  How  many  inhabitants  in  these  9  cities  of 
Ohio  ? 

9.  Three  farmers,  last  year,  sold,  fruit  as  follows  :  The  first, 
79  bushels  of  apples,  391  bu.  of  peaches,  8  bu.  of  quinces,  39  bu. 
of  pears,  and  8  bu.  of  cherries.  The  second,  43  bu.  of  apples,  539 
bu.  of  peaches,  19  bu.  of  quinces,  37  bu.  of  pears,  47  bu.  of  plums, 
and  6  bu.  of  cherries.  The  third,  27  bu.  of  apples,  and  87  bu.  of 
peaches.  (1)  How  many  bushels  of  each  kind  of  fruit  were  sold  ? 
(2)  How  many  bushels  of  fruit  did  each  farmer  sell  ?  (3)  How 
many  bushels  did  all  of  them  sell  ? 

10.  North  America  is  inhabited  by  54,566,936  people ;  the 
West  Indies  by  4,316,718  ;  South  America  by  26,913,531 ;  Europe 
by  311,694,029;  Asia  by  791,031,473;  Africa  by  199,921,600; 
Oceanica  by  38,318,771.     Find  how  many  in  all.    (Statistics  1885.) 

11.  Three  farmers  divided  their  land  as  follows  :  The  first  had 
6  acres  in  rye,  20  acres  in  wheat,  69  acres  in  corn,  2  acres  in  pota- 
toes, 26  acres  in  meadow  land,  19  acres  were  lying  fallow,  and  3 
acres  were  in  grapes.  The  second  had  12,  39,  136,  5,  75,  26,  2 
acres  (take  them  in  the  same  order).  The  third  had  4,  19,  53, 19, 
5,  4,  9.  (1)  Find.,  how  many  acres  of  each  kind  ;  (2)  how  many 
acres  to  each  farmer  ;  (3)  How  many  acres  in  all. 

12.  Lake  Superior  has  an  area  of  32,000  square  miles ;  Lake 
Michigan  24,000,  Lake  Huron  20,400,  Lake  Erie  9,600,  Lake 
Ontario  6,300.     What  is  the  total  area  of  the  five  great  lakes  ? 


34:  STANDARD  ARITHMETIC. 

Original  Problems. 

28a  Note. — Problems  such  as  the  first  may  be  made  up  under  the  direction  of 
the  teacher  with  the  aid  of  all  the  pupils  of  the  class,  due  notice  having  been  given 
of  the  kind  of  contributions  desired.  One  problem  per  day,  if  possible,  should  be 
required  of  each  pupil. 

1.  Find  the  number  of  pages  read  by  all  the  pupils  in  books 
not  used  in  school.  (Each  reports  for  himself,  all  set  down  the 
items  and  find  the  sum.) 

2.  Find  how  many  examples  all  have  solved  within  any  given 
time  ;  how  many  lines  all  have  read ;  how  many  words  all  have 
spelled  or  missed ;  how  many  chestnuts,  walnuts,  hickory  nuts, 
acorns,  etc.,  all  have  gathered. 

3.  In  a  rural  district  an  enumeration  may  be  made  of  the  num- 
ber of  horses,  cows,  sheep,  hogs,  chickens,  etc.  ;  of  the  apple, 
peach,  cherry  trees,  etc.  ;  of  the  eggs  gathered,  etc.,  etc.,  in  the 
district,  or  on  all  the  farms  from  which  the  children  come.  Let 
them  report  in  writing,  so  that  no  sensitive  child  or  parent  be 
offended. 

Note. — Such  problems  as  the  following  should  be  written  out  by  the  pupils 
separately,  and  without  the  aid  of  the  teacher.  Paper,  cut  to  uniform  size,  should 
be  used  for  the  purpose,  and  the  exercises  corrected,  as  in  other  language  lessons. 

4.  Each  pupil  may  imagine  himself  to  be  a  store-keeper  in  any 
line  of  trade  he  pleases,  and  make  problems  in  regard  to  it.  He 
should  write  out  each  problem  with  the  utmost  care,  and  be  sure 
that  he  knows  the  answer  before  he  gives  it  to  the  class. 

5.  Questions  may  be  prepared  with  the  aid  of  a  text-book  in 

geography ;  as,  for  instance  :  What  is  the  population  of  the 

largest  cities  of  ?  (the  pupil's  own  state,  the  United  States, 

any  foreign  country,  or  the  world). 

6.  The  teacher  can  at  sight  judge  of  the  correctness  of  answers 
to  such  questions  as  the  following  :  If  there  are  ten  wagons,  and 
each  one  contains  23  bushels  of  apples  and  27  bushels  of  potatoes, 
how  many  bushels  of  potatoes  and  how  many  bushels  of  apples  in 
them  all  ? 


CHAPTER   III. 

SI)  BTRACTION. 
Units  and  Tens. 

Examples. — l.  Fred  had  13  cents,  and  bought  a  top  costing 
80.  How  many  cents  had  he  left  ?  How  many  would  he  have 
had  left  if  the  top  had  cost  7^  ?  6^  ?  4^  ?  5^  ?  9^  ?  3^  ?  (<f  is  a 
sign  for  cents. ) 

2.  In  a  class  of  16  pupils  there  are  9  girls.  How  many  boys 
are  in  the  class  ? 

3.  Emily  is  18  years  old,  and  her  brother  is  9  years  younger. 
How  old  is  he  ?  Their  sister  is  11  years  younger  than  Emily. 
How  old  is  the  sister  ? 

4.  A  little  boy  had  9  marbles,  and  bought  enough  more  to 
make  15.     How  many  did  he  buy  ? 

5.  What  number  must  we  add  to  7  to  make  14?  16?  19? 

6.  Beginning  with  100  take  away  twos,  till  nothing  is  left. 
From  99  take  threes,  till  nothing  is  left ;  from  100  take  fours ; 
from  100  take  fives ;  from  96  take  sixes ;  from  98  take  sevens ; 
from  96  take  eights  ;  from  99  take  nines. 


Definitions. 

29.  Subtraction  is  a  process  of  finding  what  is  left  of  a  num- 
ber when  a  part  of  it  is  taken  away  ;  also,  of  finding  the  difference 
between  two  numbers. 

30.  Terms  used.— The  number  from  which  a  subtraction  is 
made  is  called  the  Minuend.     The  number  which  is  subtracted  is 


36  STANDARD  ARITHMETIC. 

called  the  Subtrahend.  What  is  left  of  a  number  after  taking 
away  a  part  of  it  is  called  the  Remainder.  The  remainder  is  the 
Difference  between  the  minuend  and  subtrahend. 

Note. — Minuend  means  to  be  diminished,  and  Subtrahend,  to  be  subtracted. 
Numbers  added  are  sometimes  called  Addends,  or,  more  properly,  Addenda. 

31.  Signs. — The  sign  —  (minus)  placed  between  two  numbers 
indicates  that  the  number  on  the  right  is  to  be  taken  from  the 
one  on  the  left ;  as,  7 — 2=5,  which  is  read,  7  minus  2  equals  5. 
Minus  means  less. 


< 

3RAL     EXERCISES. 

1  N 

TENS 

A  N  C 

I     UNITS. 

8. 

9. 

10. 

11. 

12.       •           13. 

6—2  = 

8-3: 

7-4= 

11-2  = 

12-5=          13-7= 

26-4= 

14  —  3: 

35—2  = 

41—4= 

42—3=          25—6= 

54-2  = 

46-5: 

27-6  = 

53— fc= 

76-7=         57-8= 

78-6= 

68-7: 

59—8= 

65-8= 

64—5=          63—9  = 

88—8= 

58  —  3: 

63—2= 

71—2= 

52—3=          35—7= 

92-2= 

86-5: 

75-4= 

83—4= 

26-9=          41—5  = 

14-103. 

From 

each 

of  the 

following 

numbers  subtract  3  as 

any  times  as  you  can. 

Subtract  also  5, 

7,9, 

2,  4,  and  8. 

10 

20 

•30 

40 

50 

60 

70 

80        90 

11 

21 

31 

41 

51 

61 

71 

81         91 

12 

22 

32 

42 

52 

62 

72 

82         92 

13 

23 

33 

43 

53 

63 

73 

83         93 

14 

24 

34 

44 

54 

64 

74 

84         94 

15 

25 

35 

45 

55 

65 

75 

85         95 

16 

26 

36 

46 

56 

66 

76 

86         96 

17 

27 

37 

47 

57 

67 

77 

87        97 

18 

28' 

38 

48 

58 

68 

78 

88         98 

19 

29 

39 

49 

59 

69 

79 

89         99 

104.  Tell  how  many  must  be  added  to  4  (see  line  of  numbers 
below)  to  make  7,  how  many  to  7  to  make  13  (the  next  higher 
number  ending  in  the  next  figure),  and  so  on. 

4,  7,  3,  8,  7,  9,  6,  7,  4,  3,  0,  8,  6,  9,  5,  4,  5,  8,  6,  5. 

Note.— Say  4  and  3  (=7),  7  and  6  (  =  13),  3  and  5  (=8),  8  and  9  (=17). 
Omit  numbers  in  parenthesis. 


SUBTRACTION.  37 

105.   From  each  number  following  subtract  the  sum  of  its 
digits;  thus, 

5  +  6=11,  56  —  11=45.     Announce  only  results,  11,  45. 
56,  44,  32,  53,  65,  87,  48,  29,  51,  73,  18,  92,  47,  64,  20,  70,  98,  85. 


Applications. — 106.  Mr.  Smith  is  70  years  old,  his  son  is  40. 
How  old  was  Mr.  S.  when  his  son  was  born  ? 
How  much  greater  is  7  tens  than  4  tens. 

107.  Sarah  had  30  cents,  and  her  sister  50.  How  many  more 
did  the  sister  have  than  Sarah  ? 

108.  John  has  20  marbles,  William  60.  How  many  more  has 
William  than  John  ? 

109.  There  are  52  houses  on  the  east  side  of  Linden  St.,  and 
only  20  on  the  west.  How  many  more  on  the  east  than  on  the 
west  side  ? 

110.  A  farmer  sold  32  of  his  92  sheep.  How  many  had  he  left  ? 
How  many  would  he  have  had  left  if  he  had  sold  12  more  ? 

ill.  One  book  has  84  pages,  another  72.  How  many  more 
pages  in  the  first  than  in  the  second  ? 

112.  If  you  had  55^  in  your  bank,  how  many  would  be  left  if 
you  were  to  take  out  150  ? 


ORAL    EXERCISES. 

Speak  only  the  remainders,  thus,  97,  93,  88,  etc. 

113.  100-3-4-5-7-9-3-5-5-7-2-7-5-8-3  = 

114.  99—5—2—3—8-6—5—7-8-2—3—5—6-9—5  = 

115.  95—2—8—3—2—3  —  1—4—1—3-5-7—6—5—4= 

116.  87—3—5—7—9—8—6—4-2—3-1—4-7—6-8= 

117.  98—7—3—4-7—2—1—2-5-2-8—6—9—4-6= 
118-124.  From  90,  80,  70,  60,  50,  40,  30  take  10,  take  20,  take  30. 
125-131.  From  67,  83,  42,  56,  95,  72,  61  take  20,  take  30,  take  40. 
132-133.  From  90,  80,  70,  60,  50,  40,  30  take  16,  take  27,  take  23. 
139-145.  From  67,  87,  47,  57,  97,  77,  67  take  37,  take  17,  take  27. 


38  STANDARD  ARITHMETIC. 

In  Exercises  146-150  subtract  first  tens,  then  units.     Speak 


only  results. 

146. 

147. 

148. 

149. 

150. 

45—25= 

61—21  = 

65—35  = 

64—54= 

97-57= 

54—34= 

72—32  = 

76—46= 

75—55  = 

55_35  = 

29—19= 

83—53  = 

42—22  = 

86—46= 

58—28= 

67-37= 

94—54= 

37-17= 

97-57= 

63—33= 

73—23  = 

45-15= 

84—34= 

58—48= 

88—48= 

In  the  following  examples,  subtract  first  the  tens  then  the  units,  announcing 
only  two  results.  Thus,  in  solving  the  first,  think  30  less  10  and  say  20,  then  think 
20  less  2  and  say  18. 

153. 

31  —  12= 

42—16= 

53—18= 

64—15  = 

75-19= 

86—17= 

97-18= 

98-19= 

156-160.  Subtract  11  from  each  of  the  numbers,  21,  31,  41,  51, 
61,  71,  81,  91.     Subtract  also  13,  15,  17,  19. 

161-164.  Similarly  from  24,  34,  44,  54,  64,  74,  84,  94  subtract 
12,  14,  16,  18. 

165-168.  Similarly  from  27,  37,  47,  57,  67,  77,  87,  97  subtract 
19,  14,  18,  17. 

169-224.  Find  the  difference  between  each  number  and  the 
one  to  the  right  of  it  in  the  same  line. 


151. 

152. 

30—12= 

20—18= 

40—14= 

50—24= 

50-16  = 

40-35  = 

60—18= 

60—43= 

70-13= 

30—24= 

80—15  = 

90—42  = 

90-17= 

70—21  = 

100—19= 

100-46= 

154. 

155. 

37-19= 

62—28= 

48—29= 

73_36= 

54—28= 

84-37= 

64—27= 

95—39  = 

76—39  = 

56—48= 

88—49  = 

68—29= 

96—48= 

83-27= 

69-47= 

94-48= 

99    82 

73 

69 

58 

42 

36 

24 

91    87 

75 

63 

52 

38 

25 

12 

93    89 

69 

55 

47 

32 

21 

14 

95    81 

79 

67 

53 

41 

39 

21 

94    88 

71 

66 

57 

44 

27 

13 

99    77 

68 

56 

41 

35 

24 

17 

92    88 

77 

66 

55 

42 

37 

19 

97   85 

73 

61 

50 

45 

35 

23 

^s     ^ 

-^  ^ 

SUBTRACTION.  39 

225-230.  From  each  one  of  the  numbers  49,  52,  65,  76,  82,  93 
take  first  12,  and  from  the  remainder  take  15.  In  the  same  man- 
ner take  13  and  16  ;  14  and  17. 

231-236.  Similarly  from  41,  58,  63,  74,  85,  97  take  11,  and 
from  the  remainder  take  18 ;  also  16  and  19 ;  also  12  and  17. 


Tens  and 

Hundreds. 

237. 

238. 

239. 

240. 

200—20= 

500—50= 

300—80= 

700  —  60: 

300—70= 

600—40= 

400=40= 

800—30: 

400—50  = 

700—30= 

500—70= 

900—50: 

500—30= 

800—20= 

600—20= 

300—60; 

600—60= 

900-10= 

700—40= 

400—80: 

Subtract  the  hundreds,  then  the  tens. 

241. 

242. 

243. 

244. 

400—240= 

800—420= 

500-180= 

600—280: 

600—130= 

500—450= 

400—290= 

800—370: 

800—320= 

700—260= 

600-370= 

900—580: 

900—440= 

900—310= 

700-490= 

600—390: 

500—330= 

300—220= 

900-630= 

700—210 

Applications.— 245.  Mr.  A  has  400  sheep,  Mr.  B  200.  How 
many  has  Mr.  A  more  than  Mr.  B  ? 

246.  A  clerk  receives  a  salary  of  $1,200,  and  pays  $290  for 
rent.     How  much  remains  for  other  expenses  ? 

Note.— The  sign  %  is  used  to  denote  dollars.     It  is  called  the  dollar  mark. 

247.  Mr.  Abel  sold  his  house  for  $860.  He  had  bought  it  for 
$730.     How  much  did  he  gain  ? 

248.  If  a  boy  resides  580  steps  from  the  school-house,  but,  on 
going  to  school,  stops  on  his  way  after  taking  380  steps,  how  many 
has  he  yet  to  take  ? 

249.  A  farmer  bought  a  horse  and  two  cows  for  $195.  If  one 
cow  cost  $49,  and  the  other  $50,  how  much  did  he  pay  for  the 
horse  ? 


40 


STANDARD  ARITHMETIC. 


Units,  Tens,  and  Higher  Orders. 

32.  Case  I.  When  no  term  of  the  subtrahend  is  greater  than 
the  term  in  the  same  order  of  the  minuend. 

Example.— From  796  subtract  354. 

Illustration. — Suppose  that  Mr.  Jones  has  seven  sacks  of  money, 
each  containing  one  hundred  silver  dollars,  nine  rolls  of  ten  dol- 
lars each,  and  six  dollars  lying  loose  on  his  table,  as  represented 
in  the  following  picture,  and  that  Mr.  Smith  calls  to  collect  354 


dollars.     The  pupil  will  readily  understand  that  Mr.  Jones  has 

only  to  give  Mr.  Smith  four  of  the  single  pieces,  five  of  the  ten 

dollar  rolls,  and  three  of  the  sacks  containing  one  hundred  dollars 

each,  and  that  he  will  then  have  442  dollars  remaining. 

Note. — The  learner  should  practice  himself  in  such  illustrations  till  he  is  fa- 
miliar with  them.  He  will  thus  surely  learn  the  significance  of  the  processes  of 
arithmetic. 


SUBTRACTION.  41 

Process  of  subtraction  for  slate  work. 

Dollars.  Dollars. 

Mr.  Jones  had            7  hundreds  9  tens  6  units.  ~          796 

Of  this  he  paid          3  hundreds  5  tens  4  units.  '        354 

He  had  left                 4  hundreds  4  tens  2  units.  442 

In  the  same  way,  tell  how  you  would  take  325  from  697  but- 
tons, supposing  that  you  had  6  cards  having  100  buttons  sewed 
on  to  each,  9  cards  with  10  buttons  on  each,  and  7  loose  buttons. 
Show  how  the  remainder  would  be  found  by  work  on  the  slate. 


SLAT  E 

EXERCISES. 

1. 

2. 

3. 

4. 

5.     6. 

7. 

8. 

9. 

495 

589 

686 

798 

894    995 

789 

795 

325 

182 

273 

325 

496 

472    572 

364 

682 

124 

10. 

11. 

12. 

13. 

14. 

15. 

16. 

17. 

18. 

578 

496 

758 

983 

492 

854 

786 

854 

750 

235 

314 

245 

372 

271 

622 

543 

503 

600 

19-32.  From  879  take  213,  425,  263,  34,  728,  658,  870,  457, 
23,  43,  654,  222,  333,  400. 

33-46.  Take  432  from  each  of  the  following  numbers :  543, 
733,  645,  987,  655,  438,  679,  542,  632,  777,  989,  656,  686,  567. 


Note. — In  the  process  of  subtraction  there  are  two  modes  of  reckoning.  For 
instance,  in  subtracting  5  from  9,  some  say,  "  5  from  9  leaves  4  " ;  others,  "  5  and 
4  are  9  " ;  both  writing  the  4  as  it  is  spoken,  or  better,  as  it  comes  into  the  mind. 
The  results  are  the  same,  but  the  latter  wording  is  recommended  in  practice  for 
many  reasons,  one  of  which  is  that  it  is  less  liable  to  error.  It  should  not  be  intro- 
duced, however,  till  the  former  is  well  understood  by  the  learner. 

Example.— From  789  subtract  435. 

Wording.— 5  and  4  are  9,  3  and  5  are  8,  4  and  3  are  7.     Only 
•  ®®  the  results  printed  in  heavy  type  should  be  spoken,  and  these  should 

5  be  written  as  uttered. 

354 

This  is  called  the  "making  up  method."  Besides 
being  less  liable  to  error,  other  advantages  will  be  seen  in  Case  II. 
See  Ex.  68,  p.  44,  also  Shorter  Method  in  Long  Division,  p.  90. 


42  STANDARD  ARITHMETIC. 

33.  Case  II.  When  any  term  of  the  subtrahend  is  greater  than 
the  term  in  the  same  order  of  the  minuend. 

Example.— From  442  subtract  136. 

Illustration. — After  Mr.  Jones  had  paid  Mr.  Smith,  he  had 
$442  left,  out  of  which  he  is  paying  Mr.  Brown  $136.  But  since 
he  has  only  two  loose  dollars,  he  is  here  represented  as  having 


taken  ten  dollars  from  one  of  the  rolls,  and  put  them  with  the 
two,  thus  making  twelve  single  pieces.  From  these  he  has  put 
forward  six  pieces.  He  has  shoved  forward  also  the  three  remain- 
ing ten  dollar  rolls,  and  one  sack  containing  a  hundred  dollars. 

Thus  he  has dollars  left. 

The  process  is  represented  in  figures  as  follows  : 

Dollars.  Dollars. 

Mr.  Jones  had                   4  hundreds  4  tens  2  units.  ~           442 

Mr.  Brown  was  paid         1  hundred   3  tens  6  units.  '         136 

Mr.  Jones  had  left            3  hundreds  0  tens  6  units.  306 


SUBTRACTION. 


43 


SLATE     EXERCISES. 


Subtract 

47.         48. 

49. 

50. 

51. 

52. 

53. 

54. 

55. 

56. 

§24        543 

384 

792 

325 

986 

847 

761 

568 

738 

116        327 

259 

368 

217 

509 

719 

234 

139  ■ 

309 

57. 

58. 

59. 

60. 

61. 

62. 

63. 

64. 

65. 

66. 

382 

44 

631 

52 

166 

240 

874 

760 

293 

871 

279 

25 

408 

39 

39 

19 

65 

347 

175 

257 

Note. — The  pupil  does  not  understand  the  foregoing  illustrations  if  ho  can  not 
go  farther  and  explain  the  case,  where  he  has  to  "  borrow,"  both  from  the  tens  and 
from  the  hundreds;  or,  where  there  are  no  tens,  perhaps  also  no  units. 


ioo       ■ 

PENCILS      | 

"*i00 

pencils 

100 

PENCILS 

HUM** 

100 

PENCIL?  1 

100 

PENCILS 

too 

PENCILS 

100 

PENCILS  \ 

67.  Explain  how  you  could  most  conveniently  take  585  pencils 
from  the  number  of  pencils  represented  above. 

68.  In  the  same  way  explain  how  you  would  proceed  to  take 
378  match-sticks  from  the  number  represented  below  - 


Arithmetical  process  : 

There  are  6  hundreds,  0  tens,  4  units. 

We  take  3  hundreds,  7  tens,  8  units. 

There  will  be  left        2  hundreds,  2  tens,  6  units. 


Or, 


604 
378 
226 


44 


STANDARD  ARITHMETIC. 


According  to  the  "making  up  method"  mentioned  in  note 
at  foot  of  p.  41,  the  wording  in  the  process  of  subtracting  378 
from  604  would  be  as  follows  : 

fi~.  Eight  and  6  are  14,  carry  one  to  7,  8  and  2  are  10,  carry  1  to 

otro  3,  4  and  2  arc  6,  the  numbers  represented  in  heavy  type  being  the 

■ only  ones  spoken  aloud.     These  should  be  written  while  they  are 

being  pronounced. 

69-78.  Subtract  328  from  each  of  the  following  numbers  :  442, 
560,  643,  751,  962,  876,  777,  691,  564,  886. 

79-86.  Find  the  difference  between  435  and  each  of  the  fol- 
lowing numbers  :  561,  872,  960,  253,  864,  950,  762,  341. 

Subtract 
87.  88.  89.  90.  91.  92.  93.  94.  95.  96. 

624         733         521         635         742         815         740         490         600         586 
156  56  68        469        387        466         295         399        579         297 


97. 

98. 

99. 

100. 

101. 

102. 

103. 

104. 

105. 

106. 

350 

440 

560 

690 

720 

870 

980 

550 

360 

570 

123 

145 

467 

498 

445 

456 

379 

258 

177 

276 

Try 

107.  255 

108.  836 

109.  952 

110.  115 

111.  303 

112.  517 

113.  206 

114.  614 

Find 
139. 

7834 
4915 

146. 

628234 
149365 


to  solve  these  examples  orally  before  using  the  slate. 

378-108=       131.  457- 


-15  = 
-17= 
—19  = 
-14= 
-13  = 
—  19= 
—16  = 
—18= 


115.  256—150= 

116.  743—240= 

117.  428—320= 

118.  549—540= 

119.  387—180= 

120.  613—410= 

121.  309—200= 

122.  487-280  = 


123. 
124. 
125. 
126. 
127. 
128. 
129. 
130. 


the  differences 
140.  141. 

9425  6453 

3568  2575 


142. 

7542 
4634 


643-207= 
418-309= 
237-108= 
276-208= 
517—309= 
226—109= 
375-207= 


143. 

28769 
3454 


132.  756- 

133.  321- 

134.  432- 

135.  527- 

136.  621- 

137.  732- 

138.  425- 

144. 

52639 
4758 


•259  = 
■257= 
•134= 
246= 
358= 
•247= 
544= 
•247= 


145. 

43892 
5893 


147.       148. 

23456      180020 
15897     147835 


149.         150.        151. 

70000      60000001      3845 
39876     12345678     1578 


SUBTRACTION  45 

34-.  Utile. — 1.  Write  the  subtrahend  under  the  minuend,  so 
that  units  may  stand  under  units,  tens  under  tens,  hundreds  under 
hundreds,  etc. 

2.  Commence  at  the  right  hand,  and  if  possible  subtract  each 
term  of  the  subtrahend  from  the  one  above  it,  and  write  the  re- 
mainder below  in  the  same  order. 

3.  If  any  term  of  the  minuend  is  less  than  the  term  to  be  sub- 
tracted, add  ten  to  it,  keeping  in  mind  that  the  next  higher  term 
of  the  minuend  must  then  be  diminished  by  one. 

Proof. — Add  the  remainder  to  the  subtrahend.  If  the  sum  is 
equal  to  the  minuend  the  work  is  correct. 


Examples  for  Practice  and  Review. 

1.  How  many  years  have  passed  since  the  discovery  of  America 
in  1492  ?    Since  the  discovery  of  the  Mississijjpi  river  in  1541  ? 

2.  Benjamin  Franklin  died  in  1790,  aged  84  years.  In  what 
year  was  he  born  ?  How  old  is  a  person  now  who  was  born  in 
the  year  in  which  Franklin  died  ? 

3.  A  store,  valued  at  $9,050,  was  destroyed  by  fire.  What  was 
the  owner's  loss,  there  being  an  insurance  of  $6,000  ? 

4.  There  were  12,426  soldiers  in  a  fortress,  of  whom  5,855  were 
discharged.     How  many  remained  ? 

5.  A  merchant  began  business  with  goods  valued  at  $16,810. 
After  two  years  he  found  his  goods  worth  $38,430.  How  much 
had  their  value  increased  ? 


6.  The  State  of  New  York  had  5,082,871  inhabitants  in  1880  ; 
the  State  of  Illinois  had  3,077,871.  What  was  the  difference  in 
the  number  of  inhabitants  of  these  two  States  ? 

7.  Asia  is  supposed  to  have  797,000,000  inhabitants,  and  Eu- 
rope 313,834,000.  How  many  more  people  are  supposed  to  live 
in  Asia  than  in  Europe  ? 

8.  In  1880  there  were  105,541  children  attending  the  public 
schools  of  Philadelphia,  59,768  in  Boston,  and  270,176  in  New 
York.  How  many  more  in  Philadelphia  than  in ,  Boston  ?  How 
many  more  in  New  York  than  in  Philadelphia  f 


£6  STANDARD  ARITHMETIC. 

9.  Mt.  Everest,  in  Asia,  which  is  29,002  ft.  high,  is  22,709  ft. 
higher  than  Mt.  Washington,  in  New  Hampshire.  How  high  is 
Mt.  Washington  ? 

EXERCISES  FOR  SLATE  WORK. 

10.  From  750  subtract  75,  from  the  remainder  subtract  75, 
from  the  second  remainder  subtract  75,  and  continue  this  process 
of  subtraction  till  you  have  subtracted  75  ten  times. 

11.  From  4,590  take  459  ten  times,  as  in  the  preceding  example. 

12.  From  6,380  take  638  nine  times.  Before  doing  the  work, 
say  what  the  last  remainder  will  be. 

13.  Write  any  number  expressed  by  three  figures,  annex  a 
cipher,  and  from  the  number  thus  formed  take  the  first  one  ten 
times.     Why  is  the  answer  the  same  as  in  10-11  ? 

14.  Copy  and  solve  example  237  (page  25) ;  add  the  first, 
second,  and  third  columns  separately ;  subtract  the  sum  of  the 
second  column  from  the  sum  of  the  third.  Why  should  the 
remainder  be  equal  to  the  sum  of  the  first  column  ? 

Suggestion. — There  will  be  no  difficulty  in  stating  why,  if  you  lay  out  before 
you  numbers  of  objects  as  represented  in  any  one  of  the  examples. 

15.  Copy  and  perform  examples  118-121  (page  21),  and  sub- 
tract the  sum  of  the  first  column  from  the  sum  of  the  third. 
Why  should  the  difference  be  equal  to  the  sum  of  the  second  ? 

Find  the  differences : 
16.  7432     17.  8397     18.  9465     19.  7546     20.  4932     21.  5432     22.  6420 
2345  4567  7656  1667  2784  3765  2574 

23.  4923     24.  6843     25.  7110     26.  8435     27.  9425     28.  4620     29.  7005 
4486  1876  3465  3583  2754  1629  1967 

30.  9000     31.  50,000     32.  60,000     33.  90,000     34.  80,000     35.  1,000,000 
5793  14,312  24,635  45,678  39,876  493,624 

36.  200,000       37.  700,000       38.  8,000,000       39.  1,000,000       40.  654,321 
146,231  102,009  4,563,921  912,345  200,000 


SUBTRACTION.  47 

41.  423,021  42.  524,632   43.  635,124   44.  543,210   45.  74,321,000 

156,798  213,738       78,987      244,567      56,543,289 

46.  5,246,312  47.  342,151   48.  624,001   49.  632,031   50.  73,500,493 

1,472,536  147,367      175,548      234,567      12,345,678 

51.  43,821  52.  54,312  53.  64,213  54.  75,314  55.  86,753  56.  840,170 

34,547  34,343     23,456     62,345     23,542     654,398 

57.  724,314  58.  842,531   59.  904,030   60.  800,012   61.  60,012,345 

342,675  123,654      654,321      187,654      45,678,987 

62.  8,421,303  63.  4,621,621   64.  725,321   65.  372,100   66.  743,628 

3,544,534  26,562      46,845      193,876      100,000 

67.  7,432,100  68.  85,731,465    69.  74,000,321    70.  9,333,122,210 

2,876,201  74,635,679       49,898,767       7,457,935,767 


Miscellaneous  Examples. 

Addition  and  Subtraction. 

1.  If  the  sum  of  two  numbers  is  36,251,  and  the  greater  one 
is  26,659,  what  is  the  smaller  number  ? 

2.  From  the  sum  of  3,742  and  89,331,  take  the  sum  of  1,137, 
2,065,  and  3,820. 

3.  Add  74,321,  85,746,  25,100,  321,098;  subtract  26,304,  and 
from  the  remainder  take  54,876. 

4.  Add  the  difference  between  4,321  and  3,571  to  the  difference 
between  52,312  and  19,936. 

5.  From  the  difference  between  533,016  and  154,693,  subtract 
the  difference  between  19,876  and  17,987. 

6.  To  what  number  must  893  be  added  four  times  to  make 
3,804  ? 

7.  From  what  number  must  309  be  subtracted  five  times  to 
leave  173  ? 

8.  From  what  number  must  you  take  8,763  to  leave  3,849  ? 

9.  To  what  number  must  you  add  89,650  to  make  108,731  ? 


48  STANDARD  ARITHMETIC. 

10.  How  many  times  must  038  be  taken  from  7,280  to  leave 
900  ?  to  leave  262  ?  

11.  Fred,  had  143  marbles.  How  many  had  he  left  after  giving 
19  to  you,  25  to  me,  38  to  Paul,  49  to  Edward,  and  losing  3  ? 

12.  A  congregation  had  raised  $78,596  for  the  erection  of  a 
building  which  was  to  cost  $125,000.  How  much  yet  remained 
to  be  raised  ? 

13.  A  florist  had  3,746  tuberose  bulbs.  How  many  were  left 
after  selling  815,  150,  387,  479,  and  1,091  ? 

14.  After  a  robbery  a  banker  finds  $1,657  in  his  safe.  The 
evening  before  he  had  left  in  it  $9,336.  How  much  had  been 
stolen  ? 

15.  In  the  year  1880  Chicago  had  503,185  inhabitants,  Cincin- 
nati 225,139,  St.  Louis  350,518.  1.  Find  the  sum.  2.  How 
many  had  Chicago  more  than  St.  Louis  ?  3.  More  than  Cincin- 
nati ?    4.  How  many  had  St.  Louis  more  than  Cincinnati  ? 


The  distance  by  rail  from  New  York 

To  Albnny  is  142  miles.  To  Chicago, '  977  miles. 

"  Buffalo,      439    .*«  "  Bloomington,  1,104      " 

"  Cleveland,  622     "  "  Jacksonville,   1,193    " 

"  Toledo,       735     "  "  St.  Louis,         1,285     " 

16.  By  the  aid  of  this  table  reckon  the  distance  of  Albany 
from  each  place  named  after  it. 

17.  Also  reckon  the  distance  from  Cleveland  to  Buffalo ;  to 
Chicago ;  to  St.  Louis.  Also  from  St.  Louis  to  Chicago  ;  from 
Chicago  to  Buffalo. 

Suggestion. — Write  out  a  tabic,  showing  the  distance  from  each  city  named  in 
the  foregoing  list  to  the  next,  Make  other  problems  from  this  or  other  tables  of 
the  kind. 

18.  A  farmer  had  in  his  yard  31  chickens,  17  geese,  24  turkeys, 
and  these,  with  his  ducks,  made  up  the  entire  number  of  his 
poultry,  which  was  97.     How  many  ducks  had  he  ? 


SUBTRACTION.  49 

19.  How  many  times  can  93  yds.  be  cut  from  a  piece  of  twine 
385  yds.  long  ?    How  much  will  be  left  ? 

20.  A  train  started  with  374  passengers.  At  the  first  station 
16  left  and  9  got  on  ;  at  the  second  11  left  and  25  got  on  ;  at  the 
third  3  left.  How  many  passengers  were  on  the  train  as  it  entered 
the  fourth  station  ? 

21.  In  the  six  working  days  of  a  week  a  newsboy  bought  76 
papers  a  day,  except  Friday  and  Saturday,  when  he  bought  10 
more  each  day.  He  sold  all  but  3  on  Monday,  and  4  on  Saturday. 
How  many  did  he  sell  that  week  ? 

22.  Bought  a  pair  of  ponies  for  $158,  and  sold  them  so  as  to 
gain  $47.     What  did  I  sell  them  for  ? 

23.  Bought  one  pony  for  $100,  and  another  for  $76  ;  paid  $2 
for  shoeing  each  of  them,  and  sold  the  pair  for  $210.  How  much 
did  I  gain  ? 

24.  A  farmer  has  in  one  lot  53  beech,  87  maple,  18  hickory, 
54  walnut,  28  poplar,  and  327  oak  trees.  He  sells  all  the  walnut, 
13  hickory,  78  maple,  and  15  poplar  trees  for  lumber.  He  cuts 
281  oaks  into  railroad  ties,  and  all  the  remaining  oak  and  other 
trees  for  firewood.     How  many  does  he  cut  for  firewood  ? 

25.  Bought  a  horse  and  carriage  for  $428,  and  in  selling  them 
shortly  afterward  lost  $35  on  the  carriage,  but  gained  $16  on  the 
horse.     How  much  did  I  sell  them  for  ? 

26.  On  Monday  Robert  finds  43  eggs,  25  each  on  Tuesday  and 
Wednesday,  26  on  Thursday,  22  on  Friday,  26  on  Saturday.  The 
next  week  he  finds  as  many  and  17  more.  He  sells  96  to  a 
neighbor,  and  the  cook  uses  58.  How  many  has  he  to  send  to 
market  at  the  end  of  the  two  weeks  ? 

27.  The  number  of  days  in  each  month  of  the  year  is  : 

January,    31.  April,  30.  July,  31.  October,      31. 

February,  28.  May,    31.  August,        31.         November,  30. 

March,       31.         June,  30.         September,  30.         December,  31. 

How  many  more  days  in  the  last  6  than  in  the  first  6  months  ? 


50  STANDARD  ARITHMETIC. 

28.  How  many  days  in  all  the  months  which  have  a  for  the 
second  letter  of  their  names  ?    In  all  the  rest  of  the  year  ? 

29.  How  many  days  in  all  the  months  which  have  the  letter 
c  in  their  names  ?    In  all  the  rest  of  the  year  ? 


Original  Problems. 
35.  Write  problems  for  yourself  and  classmates. 

Note. — The  following  skeleton  problems  may  be  used  at  first,  if  thought  best. 

1.  goes  to  the  store,  buys  —  for  — fs  and  —  for  — $ 

How  much  change  does bring  home  out  of  — <p  ? 

2.  has  — <fi,  buys  — ^  worth  of  — ,  and  —  yards  of  tape, 

but  forgets  what  she  paid  for  it.     She  has*  — $  left,  and  has  lost 
nothing.     What  did  she  pay  for  the  tape  ? 

3.  wished  to  buy  — ,  which  would  cost  — f ;  had  saved 

up  —  $,  and  uncle  would  give  — <p.     How  much  more  was  still 
needed  ? 

4.  , ,  (three  ladies),  buy  —  bolts  of  muslin,  each 

containing  39  yards.     Mrs.  takes  —  yards,  Mrs. 

yards,  Mrs. yards.     They  send  the  rest  to  a  poor  neigh- 
bor.    How  many  yards  does  she  receive  ? 

5.  has  —  inhabitants ;   has  not  so  many  by  — . 

How  many  has  the  latter  ? 

Hints. — Boys  raise  money  for  a  foot  ball ;  make  contributions  for  buying  an 
overcoat  or  pair  of  shoes  for  a  poor  schoolmate;  cut  ties  for  a  railroad;  buy  and 
sell  newspapers.  The  girls  make  squares  for  a  quilt ;  cakes  for  a  picnic.  Father 
gets  and  pays  out  money.  Compare  the  heights  of  mountains,  distances  of  cities 
from  each  other,  of  places  from  the  school-house,  of  the  weight  of  a  dozen  boys 
with  that  of  as  many  girls  of  about  the  same  age,  each  giving  his  or  her  cwn 
weight.    Ask  parents  for  problems. 


CHAPTER    IV. 

MULTIPLICATION. 

1.  How  many  marks  are  there  here  ?     (Count  by  3's.) 

///  ///  ///  in  in  in  in  in  in 

2.  How  much  will  6  tops  cost  at  4^  a  piece  ? 

Suggestion. — If  you  had  learned  nothing  more  of  arithmetic  than  how  to  count 
by  4's,  you  could  take  one  top  and  pay  for  it,  and  then  another  and  pay  for  that, 
and  so  on  till  you  had  six  tops,  and  had  put  down  six  piles  of  4^  each  to  pay  for 
them.     Then,  counting  by  fours,  you  would  find  24^  to  be  the  cost  of  the  six  tops. 

3.  In  like  manner  with  pebbles,  acorns,  grains  of  wheat,  match- 
sticks,  marks  upon  the  slate,  or  other  convenient  objects,  for 
counters,  find  the  cost  of  4  oranges  at  7^  a  piece  ;  also,  separately, 
of  6,  8,  and  9  oranges  at  5^  a  piece. 

4.  In  the  same  way  illustrate  how  you  might  find  the  cost  of  9 
pounds  of  sugar  at  5<fi,  at  7^,  at  10^,  at  8$,  at  6^,  at  9^  a  pound. 

36.  The  process  of  counting  in  this  way  would  become,  in  a 
short  time,  very  tiresome.  It  would  certainly  be  more  convenient 
to  learn,  once  for  all,  the  sum  of  five  8's,  than  to  have  to  find  the 
sum  every  time  we  need  to  know  it. 

Finding  the  sum  of  8  +  8  -f-  8  4-  8  4-  8,  as  we  have  already  learned, 
is  called  Addition,  but  taking  40  for  5  times  8,  without  at  the 
same  time  making  the  addition,  is  Multiplication. 

Having  constructed  a  table  which  shows  the  sum  of  from  one 
to  ten  l's,  2's,  3's,  4's,  etc.  (see  Ex.  10,  p.  20),  the  pupil  should 
now  commit  it  thoroughly  to  memory. 

For  convenience  it  is  here  given  in  full,  with  the  addition  of 
the  ll's  and  12's„ 


52 


STANDARD  ARITHMETIC. 


Multiplication   Table. 


9   I  10  I  II   I  12 


24  |  36  |  48  Wllm  72  J  84  J  96  1 1  Q8|ff|l  1  32 


Directions  for  learning  the  Multiplication  Table. — 1.  Multiply  2  (see  figure 
at  the  head  of  the  second  «olumn)  by  the  numbers  in  the  first  column,  repeating 
the  two  numbers  and  their  product,  thus :  2  times  2  are  four,  S  times  2  are  6,  4 
times  2  are  8,  5  times  2  are  10,  etc.  After  learning  a  column  in  this  manner,  it 
will  be  well  to  reverse  the  order,  and  learn  2  times  2  are  4,  2  times  3  are  6,  2  times 
4  are  8,  etc. 

2.  The  shading  indicates  the  parts  which  are  learned  most  easily.  The  real 
difficulties  of  the  table  are  to  be  found  in  the  unshaded  parts.  Special  attention 
should  therefore  be  giyen  to  them. 


Definitions. 

37.  Multiplication  is  a  short  process  of  finding  the  sum  of 
two  or  more  equal  numbers. 

38.  Terms  used. — When  we  say  9  times  3=27,  we  multiply 
3  by  9.  The  numbers  that  are  multiplied  together  are  called 
the  Factors  (makers)  of  the  result. 

Note. — Think  of  the  word  factory,  a  place  where  things  are  made. 

39.  The  factor  which  is  multiplied  is  called  the  Multiplicand. 
The  factor  which  we  multiply  by  is  called  the  Multiplier.     The 


MULTIPLICATION. 


53 


result  obtained,  that  is,  what  is  produced  by  multiplication,  is 
called  the  Product  (the  thing  produced). 

Note. — Twice  a  number  is  double  that  number,  three  times  is  triple,  four  times 
is  quadruple,  etc.  Any  number  of  times  another  number  is  a  multiple  of  (that  is, 
many  times)  the  number.  Hence,  any  product  of  a  number  is  a  multiple  of  that 
number.     The  multiplication  table  is  a  table  of  multiples. 

Notice  -multi  in  multiply,  multiplicand,  multiplier,  multiple.  Multi  means  mam/, 
and  ply,  pli,  or  pie,  means  fold.     Multiplier  means  many  folder. 

40 ■  Signs. — The  sign  X  is  used  to  show  that  two  numbers 
are  to  be  multiplied  together,  as  4x7=28  may  be  read,  4  multi- 
plied by  7  equals  28,  or  4  times  7=28.  The  latter  reading  will 
be  preferred  in  this  book. 

The  product  of  two  factors  is  the  same,  whichever  is  used  as 
the  multiplier ;  hence,  in  performing  a  multiplication,  we  gener- 
ally use  the  one  containing  the  fewest  significant  figures,  because 
it  is  most  convenient  to  do  so ;  but,  in  indicating  a  multiplica- 
tion, it  is  best  for  the  learner  to  write  that  term  before  the  sign 
( X )  which  properly  comes  before  the  word  times  in  stating  the 
reason  for  the  multiplication. 


SLATE     EXERCISES. 

5.  After  learning  the  table  on  page  52,  complete  the  following 
table.     Make  other  tables,  changing  the  order  of  the  factors. 


7 

2 

8 

5 

3 

9 

1 

6 

10 

10 

70 

20 

80 

50 

30 

90 

10 

60 

100 

6 

42 

12 

7 

9 

3 

5 

8 

* 

54 


STANDARD  ARITHMETIC. 


ORAL     EXERCISES. 

6.  Write  in  order,  5,  7,  4,  1,  8,  3,  6,  9,  2,  10,  and  multiply 
each  number  by  2,  by  3,  by  4,  etc.,  to  10. 

Caution. — Do  not  say,  3  times  5  are  15,  3  times  7  are  21,  etc.,  but,  pointing  at 
5,  *7,  4,  etc.,  and  knowing  that  you  are  to  multiply  each  by  3,  say  15,  21,  12,  etc. 

7.  Write  the  following  lines  of  figures  upon  slate  or  paper  : 

4,  7,  2,  8,  8,  5,  6,  3,  9,  2,  6,  6,  8,  7,  9,  5,  3,  3,  4,  2,  5, 

5,  3,  7,  7,  5,  5,  4,  6,  9,  9,  8,  4,  4,  2,  3,  8,  5,  4,  9,  6,  7. 

Then,  pointing  successively  between  4  and  7,  7  and  2,  2  and 
8,  etc.,  announce  their  products,  thus  :  28,  14,  16,  etc. 

Note. — This  affords  an  excellent  exercise  upon  the  multiplication  table,  within 
10  times  10,  and  when  the  pupil  can  give  the  products  almost  as  rapidly  as  he  can 
speak,  he  is  ready  to  go  forward,  and  not  till  then. 


Seven 
times 

T  X  live  tens  are  thirty-five  tens. 

Tens.— 8.  Multiply  90,  30,  60,  50,  80,  20,  70,  40,  by  1 ;  2  ;  3 ; 
4  ;  5  ;  6  ;  7  ;  8  ;  9. 

Note.— Here   the   pupil   should   Hunk  4  times  9  tens,  but  should  say  only, 
"36  tens,  or  360";  "12  tens,  or  120,"  etc. 


Seven 
times 


7  x  five  hundred  are  thirty-five  hundred. 

Hundreds.— 9.  Multiply  600,  300,  900,  200,  800,  500,  400,  by 
9;  3;  6;  8;  5;  2;  4;  7,  and  give  the  results,  thus  :  54  hundred, 
or,  5  thousand  4  hundred. 

Thousands.— 10.  Multiply  7000,  5000,  8000,  2000,  9000,  3000, 
6000,  by  8  ;  5  ;  3  ;  9  ;  4 ;  7  ;  2  ;  6. 


Wc  add  thus:  "5,  10,  15,  etc.,  to  45."  Then  setting  down 
the  5  units,  and  carrying  the  4  tens  to  the  column  of  tens,  we 
say:  "4,  12,  20,  etc.,  to  76,"  and  set  down  the  six  tens,  carry  the 
7  hundreds  to  the  next  column,  and  add  as  before.     Or, 


MUL  TIP LIC A  TION.  55 

SLATE     WORK. 

41.  Units,  Tens,  Hundreds,  etc.— n.  Write  4685        Addition. 

nine  times,  as  in  the  margin,  and  add.  4685 

4685 
4685 
4685 
4685 

4685 
Since  all  the  figures  in  each  column  are  the         4685 

same,  we  save  time  by  taking  45  at  once  for  "9         4685 

times  5,"  as  learned  in  the  multiplication  table.    We         4685 

then  set  down  5,  and  carry  4  to  9  times  8,  and  so  on,        42f65 

exactly  as  if  we  were  finding  the  sum  by  addition. 

Although  4685  is  here  written  9  times,  as  in  addition,  the 
last  process  by  which  we  obtain  the  result  is  Multiplication. 

In  multiplying,  however,  we  save  more  than  the 
time  required  for  counting  by  5's,  by  8's,  etc.     We      Multiplication, 
save  writing  more  than  once  the  number  to  be  mul-         4685 

tiplied,  by  simply  noting  the  number  of  times  each        - 

term  is  to  be  taken,  as.  in  the  margin. 

Wording. — Knowing  that  you  are  to  multiply  5  by  9,  do  not  repeat  "  9  times  5 
are  45,"  but  say  at  once  "45,"  and  write  5  in  units'  place  as  the  word  is  pro- 
nounced. Then  say  72,  76,  not  "  9  times  8  are  72  and  4  are  76."  Use  as  few 
words  as  possible. 

Examples.— 12.  What  is  the  value  of  9  acres  of  land  at  $783 
per  acre  ? 

13.  At  $16856  per  mile,  what  is  the  cost  of  constructing  8 
miles  of  railway  ? 

14.  What  will  6  horses  cost  at  $273  each  ? 

15.  What  is  the  cost  of  building  7  locomotives  at  $13586  apiece  ? 

16.  What  is  the  value  of  5  pianos  at  $785  each  ? 

17-100.  Multiply  each  of  the  following  numbers  by  6  ;  by  9 ; 
by  3  ;  by  7  ;  by  4  ;  and  by  8  : 

4759    5678    2134    4157    5132  5426    3627 

3846    8679    7986    9768    8979  6978    7896 


56 


STANDARD  ARITHMETIC. 


Multiplying  by  10,   100,  1000,  etc. 

101.  Write  367  ten  times  and  add.  How  many  times  367  are 
there  in  the  sum?     How  do  the  figures  in 

the  sum  differ  from  the  figures  in  367  ? 

102.  Write  3670  ten  times  and  add.  How 
many  times  367  in  36700  ?  How  do  the  fig- 
ures of  this  sum  differ  from  those  of  367  ? 

103.  Write  36700  ten  times  and  add.  How 
many  times  367  in  each  36700  ?  How  many 
times  367  in  367000  ?  How  do  the  figures  of 
this  last  sum  differ  from  those  of  367  ? 

Note. — The  results  in  the  last  three  examples  indicate 
a  principle  which  it  is  desirable  the  pupil  should  discover 
for  himself.  Additional  examples  of  the  kind  may  be  dic- 
tated by  the  teacher. 

42.  The  solution  of  the  foregoing  exam- 
ples will  enable  the  pupil  to  see  that  annexing 
one  0,  thus  removing  the  digits  of  a  number 
one  place  to  the  left,  increases  the  number 
tenfold  ;  annexing  two  O's  to  a  number,  thus 
removing  its  digits  two  places  to  the  left,  in- 
creases, or  multiplies,  its  value  ten  times  ten- 
fold, or  a  hundredfold  ;  that  annexing  three 
O's  multiplies  the  value  of  a  number  a  thousandfold,  etc. 

Note  on  the  Illustration. — One  stick 
being  taken  from  each  of  the  right  hand 
boxes  represented  above,  we  have  ten. 
These  being  tied  together,  are  placed  in 
the  tens  box  below.  Repeating  this  op- 
ration  with  the  remaining  single  sticks, 

and  proceeding  similarly  with  the  bundles  of  ten,  it  becomes  evident  to  the  senses 
that  ten  times  53  are  equal  to  530. 

Hence  we  have  the  following 

43.  Rule.— To  multiply  any  number  by  10,  100,  1000,  etc., 
annex  to  the  number  $o  be  multiplied  as  many  O's  as  there  are  O's 
in  the  multiplier. 


MULTIPLICATION.  57 

ORAL    EXERCISES 

104.  What  will  be  the  figures  of  the  results  if  you  increase 
each  of  the  following,  numbers  tenfold  :  18,  92,  37,  802,  460  ? 

105.  Multiply  3562,  8921,  7643,  284,  39,  689,  9876,  by  10. 

106.  Multiply  632,  54,  723,  140,  29,  3572,  60,  932,  807,  by  100. 

107.  Multiply  15,  269,  387,  4,  5467,  198,  3287,  6420,  by  1,000. 

108.  Multiply  6,  16,  26,  328,  10,  400,  632,  84730,  by  10,000. 

109.  Multiply  71,  83,  94,  738,  8010,  4283,  738,  by  100,000. 


Multiplying  by  any  Number  of  Tens,  Hundreds,  etc. 

44.  no.  Copy  //////    7/// //  ten  times,  and  show  that 
10  times  2X7  =  20X7.    Copy  ////  //     //////     7///// 

ten  times,  and  show  that  10  times  3  X  7  =  30  X  7. 

ill.  If  you  were  to  copy  #&  /  //// /  7/&  /  7//// 

one  hundred  times,  you  could  show  in  like  manner  that  100  times 
4  X  6  =  400  X  6. 

112.  How  many  times  five  marks  are  represented  here  ?    How 

many  marks  9    How  many  times  5  marks  in  10  such  rows  ?    How 
many  marks  t 

In  like  manner  tell  how  many  times  5  marks  in  100  rows,  and 
how  many  marks. 

Suggestion. — Let  similar  exercises  be  continued  till  the  pupil  becomes  so  en- 
tirely familiar  with  the  results  that  he  can  anticipate  them  with  confidence. 


113.  Copy  the  following,  and  add  by  lines  and  columns. 
16  +  16  +  16  +  16  +  16  +  16  +  16  +  10  +  16  +  16 
16  +  16  +  16  +  16  +  16  +  16  +  16  +  16  +  16  +  16 
16  +  16  +  16  +  16  +  16  +  16  +  16  +  16  +  16  +  16 

How  many  16's  in  each  column  ?    Three  times  16  =  ?     How 
many  columns  are  there  ?    How  many  times  3  X  16  ?    Ten  times 


58  STANDARD  ARITHMETIC. 

3  X  1G  =  ?    Ten  times  three  16's  are  how  many  16's  ?    Thirty 
times  16  then  =  ? 

What  difference  between  the  figures  in  the  products  of  3  X  16 
and  30  X  16  ?  When  you  have  found  the  product  of  3  X  16, 
how  can  you  most  readily  form  the  product  of  30  X  16  ? 


114.  If  you  were  to  write  three  lines,  each  line  containing  one 
hundred  16's,  and  were  to  add  the  columns  as  before,  the  sum 
of  each  would  of  course  be  three  times  16,  and  would  =  ?  How 
many  columns  would  there  be  ?  How  many  times  3  X  16  ?  One 
hundred  times  3  X  16  =  ?  One  hundred  times  3  X  16  are  how 
many  times  16  ?    Three  hundred  times  16  =  ? 

What  difference  between  the  figures  of  the  products  of  3  X  16 
and  300  times  16  ?  When  you  know  how  many  3  X  16  are,  how 
can  you  find  300  times  16  with  least  labor  ? 


115.  Copy  the  following  line  five  times,  setting  the  numbers 
carefully  one  under  another,  and  add  by  lines  and  columns ;  then 
question  yourself  in  the  same  manner  as  above. 

827  +  327  +  327  +  327  +  327  +  327  +  327  +  327  +  327  +  327 
Caution. — The  learner  should  not  neglect  to  perform  all  additions  as  directed. 

116.  Suppose  you  were  to  extend  the  five  lines  till  you  had  a 
hundred  327's  in  each  line  ? 

Note. — You  need  not  write  them  so  often,  but  you  may  think  of  them  as  if 
they  were  written,  and  ask  yourself, 

How  many  327's  in  a  column  ?  Five  times  327  =  ?  How 
many  columns  would  there  be  ?  A  hundred  times  five  times 
327  =  ?  What  is  the  difference  between  the  figures  of  the  prod- 
uct of  5  X  322  and  of  500  X  327  ?    Hence  we  have  the  following 

4-5«  Mule.—  When  the  multiplier  is  expressed  by  a  significant 
figure  with  ciphers  annexed,  multiply  by  the  significant  figure  and 
annex  as  many  ciphers  to  the  product  as  there  are  ciphers  at  the 
right  of  the  significant  figure. 


MUL  TIPLICA  TIOX.  59 

ORAL    EXERCISES. 


117. 

118. 

119. 

120. 

121. 

122. 

20x8= 

30x9= 

60x3= 

80x5  = 

40x6  = 

-90x4= 

20x5= 

30x4= 

60x7= 

80x8= 

40x8= 

90x9= 

20x4= 

30x8= 

60x5= 

80x3  = 

40x5= 

90x3  = 

20x7= 

30  x  7= 

60x6= 

80x9= 

40x7= 

90x7= 

20x3  = 

30x3  = 

60x4= 

80x4= 

40x3= 

90x5= 

123-130.  Multiply  5,  9,  1,  8,  2,  7,  6,  4,  3,  by  200 ;  500 ;  700 ; 
900  ;  400.     Also  by  3000 ;  8000 ;  6000. 

Note. — It  may  be  better  that  the  learner  should  not  attempt  the  following  exer- 
cises without  the  aid  of  the  slate,  but  if  he  does,  he  will  find  it  safest  to  multiply 
the  tens  first,  increasing  the  right-hand  term  of  that  product  by  what  he  sees  at  a 
glance  is  to  be  brought  up  (carried)  from  the  product  of  the  units. 


131. 

132. 

133. 

134. 

135. 

136. 

20  times 

60  times 

40  times 

90  times 

50  times 

70  times 

11 

32 

94 

51 

39 

65 

13 

45 

37 

96 

64 

39 

15 

68 

56 

77 

58 

86 

17 

93 

91 

60 

84 

69 

19 

74 

88 

70 

57 

31 

137-156. 

8x12  = 

8x25  = 

6x33  = 

30x26= 

20x48= 

5x19= 

9x34= 

8x19  = 

50x19= 

70x13  = 

8x24= 

5x26= 

7x14= 

40x18= 

50x17= 

9x18  = 

4x31  = 

9x12  = 

20  x  37= 

40  x  24= 

157-216.  Multiply  each  number  in  the  following  columns  by 
3;4;5;2;6;7;8;9. 

217-276.  Multiply  them  by  10  ;  20  ;  30  ;  40  ;  50  ;  60  ;  70  ;  80. 

277-336.  Also  by  200 ;  800  ;  600  ;  500  ;  and  by  3000 ;  7000  ; 
9000. 


52 

72 

67 

35 

79 

38 

89 

28 

47 

99 

44 

98 

24 

33 

46 

78 

14. 

77 

51 

25 

27 

29 

63 

48 

23 

55 

49 

42 

26 

83 

85 

41 

71 

34 

59 

75 

10 

96 

12 

87 

76 

53 

73 

92 

82 

88 

95 

61 

62 

57 

91 

18 

81 

30 

40 

97 

60 

70 

80 

90 

60  STANDARD  ARITHMETIC. 

SLATE     EXERCISES. 

337-403.  Multiply  each  number  in  the  columns  below  by  9  ; 
6;2;8;4;7;5;3. 

409-480.  Also  by  30  ;  50  ;  70  ;  90. 


481 

-552.  i 

ilso  by  \ 

300;  60 

0  ;  800. 

—Also  i 

by  800C 

1 ;  5000 ; 

4000. 

162 

171 

648 

207 

984 

435 

384 

613 

4212 

288 

304 

621 

318 

945 

119 

855 

533 

2345 

335 

465 

864 

424 

952 

769 

564 

828 

6789 

316 

568 

954 

154 

238 

854 

403 

261 

1234 

384 

483 

848 

925 

357 

968 

177 

695 

5678 

266 

612 

302 

193 

476 

166 

324 

785 

9123 

520 

132 

324 

965 

587 

231 

125 

674 

4567 

423 

660 

216 

164 

693 

219 

662 

753 

8912 

Note. — 1.  With  a  little  practice,  such  calculations  as  are  required  above  can  be 
performed  without  the  aid  of  the  slate.  Let  the  wording  be  the  simple  announce- 
ment of  results ;  thus, 

2.  In  multiplying  162  by  9  the  announcements  should  be  9,  144,  1458. — In 
multiplying  423  by  90  they  would  be  36,  378,  3807,  38070. 

3.  Though  these  are  excellent  exercises  for  practice,  no  learner  should  trust 
the  result  of  oral  work  in  large  numbers  without  testing  its  accuracy  by  the  written 
process.  - 

Multiplying  by  Units,  Tens,  Hundreds,  etc. 

46.  l.  Write  17  four  times  in  one  column,  ten  times  in  an- 
other, and  fourteen  times  in  a  third.    Add  the  columns  separately. 
Should  the  sum  of  the  first  two  footings  equal  the  third  ? 
Why? 

2.  Find  4  times  17  and  10  times  17  by  multiplication,  setting 
the  work  in  the  following  form  : 

4x17=  68 
10X17=170 

How  many  times  17  is  the  sum  of  the  two  products  ? 
To  find  14  times  1 7,  then,  let  the  slate-work  stand  as  follows  : 
4X17=  68 

10x17=170 

14X17=238 


MUL  TIP LIC A  TION.  61 


SLATE     EXERCISES 


In  like  manner  multiply : 

3.    3x23=  4.    5x31=  5.    4x21=  6.    8x63= 

10x23=  40x31=  20x21=  3Gx63  = 


7.    3x18= 

8.    6x71  = 

9.    4x26= 

10.    2x46  = 

20x18= 

50x71  = 

30x26= 

70x46= 

300x18= 

600x71  = 

400x26= 

200x46  = 

11.3x218=        12.     4x324=        13.     5x381=       14.     7x1236  = 

40x218=  20x324=  60x381=  20x1236= 

300x218=  200x324=  200x381=  100x1236= 

47.  The  use  of  many  unnecessary  figures  can  be  avoided  by 
writing  the  multipliers  in  their  proper  orders  and  as  one  number 
under  the  multiplicand,  as  in  the  second  form  below. 

15.  Multiply  3582  by  4376.  second  form 

3582 

FIRST   FORM.  4376 

6  times  3582=       21492  21492 

70     "     3582=     250740  250740 

300     "     3582=  1074600  1074600 

4000     "      3582=14328000  14328000 

15674832  15674832 

Though  not  nearly  as  many  figures  and  signs  are  used  in  the 
second  form  as  in  the  first,  it  must  be  remembered  that  both 
forms  mean  exactly  the  same  thing.  In  both  we  multiply  sepa- 
rately by  units,  tens,  hundreds,  etc.,  and  then  add  the  products. 
The  products  and  the  way  of  obtaining  them  are  just  the  same 
in  one  form  as  in  the  other. 


Definition. 

48.  When  a  multiplier  contains  two  or  more  terms,  the  prod- 
uct of  the  multiplicand  by  any  one  of  them  is  called  a  Partial 
product     (It  is  only  a  part  of  the  entire  product.) 


62 


STANDARD  ARITHMETIC. 


SLATE     EXERCISES. 

Arrange  the  work  in  both  forms. 
16.      3x2371=      17.      6x3582=      18.      3x1356=      19, 
40  x  2371  =  70  x  3582=  40  x  1356= 

500x2371=  300x3582=  500x1356  = 

2000  x  2371  =_      1000x3582=_      2000  x  1356= 

49.  The  positions  of  the  digits  in  each  par- 
tial product  sufficiently  indicate  their  order : 

the  ciphers  at  the  right  hand  may  therefore  be  21492 

omitted,  if  great  care  be  taken  to  set  the  first  25074 

figure  produced  by  each  multiplication  under  10746 

the  figure  by  which  you  multiply,  as  in  the  14328 

examnle.  15674832 


5x4026: 

70x4026  = 

900x4026: 

3000x4026: 

THIRD   FORM. 

3582 
4376 


20. 


Note.- 

1776 
48 


-Frequently  read  the  partial  products,  not  forgetting  the  0's. 

21.  3028        22.  4958        23.  2838        24.  3675  25.  4788 

52  74  36  49  57 


26.  6045 
65 


27.  5186 

76 


28.  2415 
35 


29.  3456 
26 


30.  2886 
47 


31.  3306 

38 


32.  2106 

33.  3569 

34.  5076 

35.  4940 

36.  5412 

37.  2401 

79 

41 

58 

65 

438 

492 

38.  3536 

39.  3436 

40.  6882 

41.  3884 

42.  1102 

43.  5605 

521 

635 

749 

362 

293 

643 

44.  Multiply  756  by  649  ;  also  by  351.  How  many  times  756 
in  the  sum  of  the  products  ? 

45-332.  Multiply  each  number  in  the  following  columns  by 
589;  by  376;  by  4863;  by  6974;  by  892;  by  3892;  by  402;  by  2009. 


706 

483 

1858 

3128 

27228 

42471 

709 

759 

6696 

5912 

59432 

60392 

245 

665 

3054 

6048 

18737 

47973 

698 

534 

2968 

3582 

29735 

46795 

596 

498 

2505 

2380 

61356 

12820 

737- 

751 

4731 

3498 

43207 

55043 

MULTIPLICATION.  63 

333-386.  Multiply  the  numbers  of  the  first  column  l)y  10  ;  by 
100 ;  by  1000.  Tell  how  you  do  it.— Multiply  them  by  30  ;  by 
400;  by  6000.  Tell  how  you  do  it.— Multiply  them  by  12;  by 
120 ;  by  1200  ;  and  tell  how  it  is  done  in  each  case. 


50.  The  following  is  the  general  rule  for  multiplication  : 

Rule  — 1.  If  the  multiplier  consists  of  one  term,  multiply  each 
term  of  the  multiplicand  by  it,  beginning-  at  the  right,  and  carry- 
ing as  in  addition.     The  result  will  be  the  product. 

2.  If  the  multiplier  contains  more  than  one  term,  multiply  sepa- 
rately by  each  term  as  above,  placing  the  right-hand  figure  of  each 
partial  product  in  the  same  order  or  column  as  the  term  by  which 
you  are  multiplying. 

3.  Add  the  partial  products  together.  The  sum  will  be  the 
entire  product. 

Proof. — Multiply  the  multiplier  by  the  multiplicand,  and  if 
the  result  is  the  same  as  before,  the  work  is  correct. 

Contractions, — If  there  are  ciphers  at  the  right  hand  of  either 
factor,  omit  them  in  multiplying,  and  annex  as  many  ciphers  to 
the  product  as  are  thus  omitted  from  both  factors. 


Examples  for  Practice  and  Review. 

1.  How  many  letters  in  a  book  of  178  pages,  if  every  page 
contains  2,978  letters  ? 

2.  The  roof  of  the  main  part  of  a  house  has  128  rows  of 
shingles,  each  row  containing  129  shingles.  How  many  shingles 
in  that  part  of  the  roof  ? 

3.  How  many  lines  are  there  in  a  work  of  15  volumes,  if  each 
volume  has  348  pages,  and  each  page  46  lines  ? 

4.  There  are  4  shelves  in  a  drug-store,  each  containing  18  jars  ; 
6  shelves,  each  containing  27  jars ;  and  5  shelves  containing  12 
jars  each.     Find  how  many  jars  in  all. 

5.  Two  school-houses  were  to  be  furnished  with  single  desks. 
One  had  18  rooms,  each  large  enough  for  52  desks ;  the  other 
had  14  rooms,  each  large  enough  for  64  desks.  Find  how  many 
desks  were  required  for  both  houses. 


64  STANDARD  ARITHMETIC. 

6.  A  contractor  ordered  52  loads  of  brick  of  1,250  each.  How 
many  bricks  did  he  order  ? 

7.  Multiply  auy  number  by  3,  by  5  and  by  2,  and  add  the 
products.  Why  is  the  sum  the  same  as  the  product  would  be  if 
the  number  were  multiplied  by  10  ? 

Multiply  as  indicated  below,  and  find  the  sum  of  each  column 
of  products.  Tell  why  the  sum  in  each  case  contains  the  same 
digits  as  the  several  multiplicands  of  the  example. 


8. 

9. 

10. 

11. 

4x327= 

5x384= 

3  x  9876= 

7x48= 

2x327= 

3x384= 

3x9876  = 

2x48= 

4x327= 

2x384= 

4x9876  = 

1x48= 

14. 

15. 

27x967= 

23  x  9807 

32x967= 

14x9807 

12x967= 

15x9807 

29x967= 

48x9807 

Perform  the  following  multiplications,  and  add  the  column  of 
products  in  each  example. 

12.  13. 

18x356=  15x438= 

19x356=  17x438= 

26x356=  49x438= 

37x356=_  19x438=_ 

Compare  the  sum  of  products  in  each  case  with  the  several 
multiplicands  of  the  same  example.  What  do  you  observe  ?  Why 
do  you  think  such  a  result  is  found  ? 

Suggestion. — Add  the  column  of  multipliers  in  each  example. 

Perform  the  following  multiplications,  add  the  products  as  in 
the  preceding  examples,  and  tell  what  you  can  about  the  results. 

16.  17.  18. 

103x587=  292x8264=  387x5379  = 

369x587=  375x8264=  203x5379  = 

287x587=  109x8264=  141x5379= 

241x587=_  224x8264=_  269x5379=_ 

Question. — Is  there  not  a  shorter  way  to  find  the  sums  of  products  as  required 
on  this  page  ? 


MULTIPLICATION.  65 

Exercises  in  Familiar  Measures. 

51.  There  are  12  inches  (in.)  in  1  foot  (ft.),  3  feet  in  1  yard  (yd.). 

Note. — If  the  pupils  are  not  already  familiar  with  these  measures,  they  should 
become  so  as  soou  as  possible.  They  should  be  required  to  measure  the  length 
and  the  height  of  desks,  the  width  of  windows,  doors ;  the  length  and  width  of  the 
room,  of  the  window-panes,  etc.,  etc.  Let  the  learner  become  familiar  with  the 
weights  and  measures  here  presented,  but  it  is  not  desirable  that  he  should  learn 
any  tables  at  this  stage,  except  by  observation  and  experience. 

19.  Find  how  many  feet  there 

are  in  46  yards.  Analysis.-Since  there  46 

are  3  ft.  in  1  yard,  there  are  3 

20.  How  many  inches  in  16,       46  times  3  a  in  46  yard8<  r^ 

79,  63,  479,  200,  571,  325  feet  ? 

21.  How  many  inches  in  one  yard  ?     In  2,  7,  9,  19,  34  yd.  ? 

22.  Find  first  how  many  feet,  then  how  many  inches,  there 
are  in  12  yards.— In  24,  36,  112,  324,  420,  500  yd. 


52.  There  are  16  ounces  (oz.)  in  a  pound  (lb.),  and  100  pounds 
in  a  hundredweight  (cwt.).     (The  c  stands  for  hundred,  and  wt  for  toeight.) 

23.  How  many  ounces  in  3,  9,  16,  32,  87  pounds  ? 

24.  How  many  ounces  in  one  hundredweight  ? — In  2,  3  cwt.  ? 

25.  How  many  pounds  in  6,  8,  3,  12  cwt.  ? 

26.  How  many  pounds  do  you  weigh  ?    How  many  ounces  ? 
How  much  would  you  be  worth  at  $125  a  pound  ? 


53.  There  are  2  pints  (pt.)   in  a  quart  (qt.),  and  4  quarts  in  a 
gallon  (gal.).     (Used  for  measuring  liquids.) 

27.  How  many  pints  are  there  in  2  quarts  ? — In  6,  8,  10,  21, 
130,  425  qt.  ? 

28.  How  many  quarts  in  6  gallons  ? — In  14,  35,  47,  81,  230, 
653  gal.  ? 

29.  How  many  pints  in  one  gallon  ?— In  2,  6,  13,  27,  35,  87. 
237  gal.  ? 

30.  How  many  pints  in  a  barrel  containing  30  gallons  ? 


66  STANDARD   ARITHMETIC. 

54.  There  are  8  quarts  (qt.)  in  a  peck  (pk.),  and  4  pecks  in  a 
bushel  (bu.).     (Used  for  measuring  fruit,  grain,  etc.) 

31.  How  many  quarts  in  2  pecks  ?— In  8, 15,  22, 49,  83, 114  pk.  ? 

32.  How  many  quarts  in  1  bushel  ?— In  7,  19,  68,  27,  346  bu.  ? 

33.  How  many  pecks  in  9,  11,  59,  90,  170,  232,  566  bu.  ? 

34.  Tell  first  how  many  pecks,  then  how  many  quarts,  there 
are  in  16,  93,  78,  136,  458  bu. 


55.  There  are  60  minutes  (min.)  in  an  hour  (h.),  24  hours  in  a 
day  (d.),  and  7  days  in  a  week  (wk.). 

35.  How  many  minutes  in  3  hours  ? — In  14,  21,  17,  23  h.  ? 

36.  How  many  minutes  in  4  h.  56  m.  ? — In  16  h.  30  m.  ? 

37.  How  many  hours  in  2  days  ? — In  16,  15,  6,  9,  43  d.  ? 

38.  How  many  hours  in  a  week  ? — In  2  wk.  3  d.  ? — In  3  wk.  2d.? 


Definitions. 


Note. — In  the  solution  of  some  succeeding  problems,  the 
learner  will  need  to  know  the  following  definitions : 

56.  The  difference  in  direction  of  two  lines 
is  an  angle. 

57.  If  the  difference  of  direction  is  such 
that  two  lines,  crossing  each  other,  make  four 
equal  angles,  the  angles  are  right  angles. 
(Right  angles  make  square  corners.) 

58.  A  rectangle  is  a  figure  having  four 
straight  sides  and  four  right  angles. 

59.  A  square  is  a  figure  that  has  four 
straight  and  equal  sides  and  four  right  angles. 

A  square  inch  is  a  square  an  inch  long  and  an  inch 
wide.     A  square  foot  is  a  foot  long  and  a  foot  wide. 


if  UL  TIP LIC A  TION. 


67 


39.  How  can  you  find  the  number  of  small  squares  in  the 
larger  one  without  counting  them 
singly?    How  many  are  there ?    How 
many  would  there  be  if  it  were  twice 
as  long  and  twice  as  wide  ? 


Note. — For  a  home  exercise  let  the  pupils 
cut  squares  of  paper  measuring  a  yard,  a  foot, 
an  inch,  on  each  side,  and  of  other  required 
sizes.  There  should  be  a  yard-stick  in  every 
school-room,  and  every  student  of  arithmetic 
should  have  a  foot-rule. 


40.  How  many  square  pieces,  each 
an  inch  long  and  an  inch  wide,  can 

be  cut  from  a  piece  of  paper  that  is  4  inches  long  and  4  wide  ? 

41.  How  many  pieces,  each  a  yard  long  and  a  yard  wide,  in  a 
piece  of  oil  cloth  6  yards  long  and  6  yards  wide  ? 

42.  How  many  small  squares  in 
this  rectangle  ?  How  many  square 
inches  would  it  contain  if  it  were 
7  inches  long  and  8  inches  wide  ? 

43.  How  many  square  feet  if  it 
were  15  feet  long  and  14  feet  wide  ? 
How  many  square  yards  if  it  were  7 
yards  long  and  3  wide  ? 

44.  How  many  square  inches  could  you  cut  from  a  piece  of 
silk  19  inches  long  and  12  inches  wide  ? 

45.  How  many  square  feet  are  contained  in  a  rectangle  13  feet 
long  and  9  feet  wide  ?    58  feet  long  and  28  feet  wide  ? 

46.  How  many  square  inches  in  a  square  foot  ?  (A  square  foot 
is  12  inches  long  and  12  inches  wide.)  How  many  in  a  square  yard  ? 
(A  square  yard  is  36  inches  long  and  36  inches  wide.) 

47.  How  many  square  feet  can  be  cut  out  of  a  newspaper 
that  is  4  feet  long  and  3  feet  wide  ?  How  many  square  inches 
could  be  cut  out  of  it  ?     (See  preceding  Example.) 


08  STANDARD  ARITHMETIC. 

Miscellaneous  Examples. 

1.  What  is  the  product  of  105,  106,  107  and  108  ? 

2.  What  must  be  subtracted  from  72x763  to  leave  34X127  ? 

3.  What  must  be  added  to  47X436  to  make  67x832  ? 

4.  Add  24X8,277  and  14x1,436,  and  from  the  sum  subtract 
8,763.     What  is  the  remainder  ? 

Perform  as  many  of  the  following  examples  as  may  be  directed, 
and  notice  the  curious  results.  Notice  the  sum  of  the  digits  in 
the  several  multipliers. 

5.  45  x  987,654,321=  6.  45  x  123,456,789  = 

7.  54x123,456,789=  8.  54x987,654,321  = 

9.  27x987,654,321=  10.  72x987,654,321  = 

11.  36x123,456,789=  12.  36x987,654,321  = 

13.  63x987,654,321=  14.  63x123,456,789= 

Note. — The  intention  here  is  not  to  teach  the  properties  of  the  numbers,  but  to 
afford  practice  in  multiplication.     The  products  may  awaken  curiosity. 

15.  From  the  sum  of  8,723,  57,  218,  9,658,  16,  and  139,  take 
the  difference  between  9,165  and  14,320,  and  multiply  the  remain- 
der by  167. 

16.  Mr.  Arnold  and  Mr.  Bayard  set  out  from  the  same  place 
to  travel  in  the  same  direction.  Mr.  A.  travels  17  m.  a  day,  and 
Mr.  B.  20  m.     How  far  apart  will  they  be  in  a  week  (6  days)  ? 

17.  If  the  minuend  is  16,  and  the  remainder  7,  what  is  the 
subtrahend  ?  If  the  minuend  is  90,087,  and  the  remainder  26,089, 
what  is  the  subtrahend  ? 

18.  Three  boys  together  buy  a  peck  of  apples  for  30^  ;  A.  pays 
9#,  B.  pays  8#,  what  does  C.  pay  ?  Four  men  are  in  partnership 
with  $16,876,  of  which  A's  share  is  $3,421,  B's  $2,500,  and  C's 
$5,693.     How  much  is  D's  share  ? 

19.  An  officer  has  an  annual  salary  of  $1,000 ;  he  spends  $75 
a  month.  How  much  does  he  save  in  one  year  ?  In  7,  9,  15,  17 
years  ? 


MULTIPLICATION.  69 

20.  A  skilled  workman  earned  $3  a  day ;  he  worked  304  days 
in  one  year.  How  much  did  he  earn  that  year  ?  He  spent  $15 
a  week  for  himself  and  family.     How  much  did  he  lay  by  ? 

21.  A  pipe  pours  into  a  reservoir  daily  13,410  gallons  of  water. 
How  many  gallons  in  30  days  ? 

22.  There  are  a  dozen  dozen  in  a  gross,  how  many  pencils  in 
75  gross  ? 

23.  A  drover  buys  12  horses  at  $85  apiece,  4  oxen  at  $68  apiece, 
35  cows  at  $56  apiece,  237  sheep  at  $4  apiece.  How  much  does 
he  pay  for  all  ? 

24.  Mr.  Ambros  sells  Mr.  Dix  516  boxes  of  soap  at  $4  a  box, 
and  buys  of  Mr.  Dix  389  bags  of  flour  at  $5  a  bag.  How  much 
does  Mr.  Dix  owe  Mr.  Ambros  on  the  transaction  ? 

25.  If  two  boys  can  dig  a  row  of  potatoes  in  an  hour,  how  long 
ought  it  to  take  one  boy  to  do  the  same  work  ?  A  piece  of  work 
can  be  done  by  2  men  in  7  days.  How  long  would  it  take  one 
man  to  do  it  ? 

26.  A  piece  of  work  which  can  be  done  in  7  days  by  45  men 
has  to  be  performed  by  one  man.     How  long  will  it  take  him  ? 

27.  There  are  24  rows  of  trees  in  an  orchard,  and  24  trees  in 
a  row,  how  many  trees  are  there  ?  If  there  were  twice  as  many 
rows  and  twice  as  many  trees  in  a  row,  how  many  trees  would 
there  be  ? 

28.  I  bought  an  overcoat,  a  vest,  and  a  hat  for  $48.  The 
overcoat  and  vest  cost  $42,  the  overcoat  and  hat  $41.  How  much 
did  each  of  the  three  articles  cost  ? 

29.  Eleven  hogsheads  of  sugar  weigh  respectively  918,  923, 
891,  .1022,  976,  889,  1019,  948,  901,  990  and  1080  pounds.  How 
much  was  the  weight  of  the  lot  less  than  it  would  have  been  if 
they  had  weighed  1086  pounds  each  ? 

30.  How  many  toes  have  238  camels,  86  ostriches,  and  453 
Canaries.  (Let  the  pupil  take  tjme  to  find  out  what  he  needs  to  know  to  perform 
this  example.) 


70  STANDARD  ARITHMETIC. 

Original  Problems. 

To  be  Composed  by  the  Pupils  for  the  Class. 

60.  l.  Give  the  prices  of  things  which  you  have  bought,  or 
which  have  been  bought  of  you,  supposing  yourself  to  be  a  sales- 
man in  a  store,  at  a  fair,  etc.,  and  ask  the  cost.  (What  must  you  give 
besides  the  prices  ?) 

2.  Problems  almost  without  number  can  be  made  about  the 
number  of  square  feet  or  yards  in  the  floors,  ceilings,  black- 
board, wall-maps,  etc.,  of  the  school-room,  giving  the  pupils  time 
to  measure  for  themselves. 

Kote. — The  pupils  proposing  such  questions  should  first  take  the  necessary 
measurements.  Let  them  take  the  nearest  whole  number  of  feet,  inches  or  yards, 
according  to  the  length  of  the  line  measured. 

3.  Give  the  number  of  rows  of  seats  and  the  number  of  grown 
people  that  can  sit  in  one  row,  and  require  the  class  to  find  how 
many  can  be  seated  in  any  church  or  hall  you  may  name. 

4.  Borrow  a  tape-line,  measure  for  yourself,  and  give  the  dis- 
tance from  one  telegraph-pole  to  another,  and  tell  the  number  of 
poles  between  any  two  places  you  may  name ;  then  ask  how  many 
feet  or  yards,  from  one  place  to  the  other. 

5.  Ask  how  many  trees  in  Mr. 's  orchard,  after  telling  the 

class  how  many  rows  there  are,  and  how  many  trees  in  a  row. 
How  many  hills  of  corn  in  a  field,  etc. 

6.  Eequire  to  know  about  how  many  apples  there  are  in  a 
wagon-load  of  50  bushels,  say  of  greenings  or  any  common  fruit 
sold  at  market.  If  the  class  can't  tell  how  it  may  be  done  with- 
out counting  all  the  apples,  tell  them. 

7.  Give  such  problems  as  these,  to  be  done  in  the  shortest 
possible  time,  changing  the  numbers  from  those  given  here  : 
What  is  the  difference  between  8  and  9  times  562  ?  35  and  45 
times  976  ?    Between  132  and  232  times  78  ?    The  sum  of  36 

times  and  64  times  84  ?     (Be  sure  that  you  see  the  point  yourself.) 


CHAPTER    V. 

DIVISION. 

1.  Write  the  letter  a  twenty-four  times  on  slate  or  paper. 
How  many  times  12  a's  are  there  in  the  24  a's?  How  many 
times  6  a's  ?    How  many  times  4  a's  ? 

2.  At  9<f  apiece,  how  many  lead-pencils  can  be  bought  for  72^  ? 
At  120  apiece?    At  8^  ?    At  6^  ? 

Suggestion. — If  a  boy  had  72  one-cent  pieces,  and  knew  no  more  of  arithmetic 
than  how  to  count,  he  could  divide  the  72  pieces  into  lots  of  90  each,  and  thus  find 
that  for  720  he  could  buy  8  tops  at  90  apiece,  or  as  many  tops  as  there  are  times 
90  in  720. 

3.  With  42  ears  of  corn,  how  many  horses  can  be  fed  if  6  ears 

are  given  to  each  ?     (Make  42  marks,  and  divide  into  groups  of  6.) 

4.  In  the  same  way,  show  how  many  dozen  there  are  in  84,  in 

60,  in  96,  in  48.      (Twelve  single  things  make  a  dozen.) 

5.  How  many  top-strings  can  be  cut  from  48  feet  of  twine,  if 
the  strings  are  made  6  ft.  long  ?  If  8  ft.  long  ? 

61.  A  thorough  knowledge  of  the  multiplication  table  super- 
sedes the  necessity  of  marks,  or  other  counters,  except  for  pur- 
poses of  illustration ;  for,  if  we  know  that  twice  12  are  24,  we 
know  equally  well  that  in  24  there  are  two  12's. 

Note  1. — For  practice  at  this  point  let  a  table,  like  the  one  suggested  on  page 
53,  be  written  on  the  black-board,  the  first  column  being  omitted.  Then  a  pupil 
pointing  successively  to  each  number  in  a  column,  and  knowing  that  the  number 
at  the  head  is  one  factor,  he  announces  the  other;  thus  under  7  he  announces  10, 
6,  7,  etc.,  as  rapidly  as  possible. 

Note  2. — Here  and  elsewhere  let  counting  be  absolutely  prohibited,  except  in 
the  way  of  illustration. 


72  STANDARD  ARITHMETIC. 

ORAL     EXERCISES. 

62.  Tens  and  Units. — Caution.  In  the  following  exercises  do 
not  repeat  3  in  18  six  times,  3  in  30  ten  times,  but  knowing  that 
you  are  to  tell  how  many  times  3  there  are  in  18,  30,  27,  etc.,  say 
at  once  6,  10,  9,  etc. 


How  many 

times 

6.  3  in 

7. 

4  in  8. 

5  in 

9.   6  in 

10. 

7  in 

11. 

8  in 

12 

.  9  in 

18 

28 

45 

54 

63 

72 

81 

30 

16 

30 

36 

35 

40 

45 

27 

32 

15 

18 

49 

56 

54 

21 

.24 

35 

42 

21 

64 

63 

24 

36 

40 

24 

56 

32 

36 

12 

48 

25 

48 

28 

48 

27 

15 

40 

50 

60 

42 

80 

72 

ILLUSTRATIVE     EXERCISES. 

Note. — The  illustrative  exercises  introduced  here  and  elsewhere  are  intended 
only  as  examples  of  what  should  be  done  in  this  direction.  Problem  after  problem 
should  be  illustrated  till  the  learner  feels  that  in  dealing  with  figures  he  is  dealing 
with  representatives  of  number,  and  that  the  arithmetical  processes  only  indicate 
what  a  person  ignorant  of  arithmetic  might  do  to  solve  similar  problems.  Let  the 
work  be  actually  performed  whenever  possible.  Labor  impresses  its  lessons  more 
deeply  than  observation.     Mere  instruction  is  not  to  be  compared  with  it. 

63.  Hundreds.— How  many  times  9  in  270? 

Write  the  letter  c  27  times  in  a  line,  and  mark  them  off  into 
groups  of  9  each  ;  thus, 

ccccccccc,  ccccccccc,  ccccccccc. 

Now,  think  how  many  c's  there  would  be  in  ten  such  lines. 
How  many  times  9  c's.  If  you  can  not  think  the  answers  to  these 
questions  without  writing  the  ten  lines,  write  them  and  count. 

64.  Thousands. — How  many  times  4  in  3600  ? 

Write  the  letter  e  36  times  in  a  line,  and  mark  them  off  into 
groups  of  4  e's  each  ;  then,  think  how  many  e's  there  would  be 
in  10  such  lines ;  in  100.  How  many  groups  of  4  in  one  line  ? 
In  10  lines  ?    In  100  lines  ? 


DIVISION-. 

ORAL  EXERCISES 

How  many 

times 

13.  3  in 

14.  6  in 

15.  7  in 

16.  8  in 

17.  9  in 

210 

480 

560 

480 

450 

240 

360 

490 

400 

630 

270 

420 

630 

720 

810 

150 

.  300 

420 

880 

540 

180 

540 

700 

640 

720 

73 


How  many  times 

18.  6  in  19.  7  in  20.  8  in  21.  9  in 

4800  1400  2400  1800 

5400  5600  8000  3600 

3000  6300  4800  6300 

1200  2800  6400  4500 


ow  many 

times 

22.  4  in 

23.  6  in 

24.  7  in 

25.  8  in 

26.  9  in 

32 

240 

2100 

6400 

450 

160 

42 

35 

5600 

63 

240 

5400 

490 

64 

8100 

3600 

480 

42 

720 

5400 

80 

4200 

6300 

80 

720 

27.  How  many  times  10  in  80?  110?  70?  120?  GO?  90?  20? 

28.  How  many  times  11  in  22?  121?  55?  132?  88?  66?  33? 

29.  How  many  times  12  in  84?  144?  72?  108?  60?  96?  132? 

Applications.— 30.  How  many  dozen  in  720?  600?  8400?  13200? 

31.  An  army  of  96000  men  is  marching  12  in  a  rank.     How 
many  ranks  are  there  ? 

32.  If  pork  is  worth  $7  a  cwt.,  how  many  cwt.  can  be  bought 
for  $6300  ?  $7700  ?  $5600  ?  840  ? 

33.  If  a  sheet  of  paper  is  folded  so  as  to  make  8  pages,  how 
many  sheets  will  be  needed  for  1600  pages  ?  5600  ? 

34.  If  a  man  walks  at  the  rate  of  4  miles  an  hour,  in  how 
many  hours  will  he  walk  48  miles  ?  3600  miles  ?  400  miles  ? 


74 


STANDARD  ARITHMETIC. 


ILLUSTRATIVE     SLATE     EXERCISES. 

65.  Thousands,  and  lower  orders. — 35.  How  many  balls  of 
twine  can  be  bought  for  $2.73,  or  273$*,  at  7^  a  ball  ?    (How  many 

times  70  are  there  in  273^.) 

Make  ten  lines  of  27  dots  each,  and  beneath  them  make  3 
single  dots.  Divide  each  full  line  of  dots  into  groups  of  seven, 
in  this  way : 


(« 


)(i 


[)(< 
->!i 


H> 


0  0 


When  the  above  work  is  neatly  done,  copy  the  following  para- 
graphs, carefully  filling  all  the  blanks  : 

1.  In  each  full  line  there  are  dots,  which  make  

groups  (of  7),  and  —  dots  over. 

2.  In  the  ten  lines  there  are  —  times  as  many  dots,  or 

dots  ;  and  —  times  as  many  groups,  or  groups ;  and  there 

are  also  —  times  as  many  over. 

3.  The  ten  6's  over  and  the  3  in  the  short  line  beneath  make 

together  —  dots.     Out  of  63  dots  we  can  make  groups  of 

7  dots  each.     Hence  there  are times  7  dots  in  273,  and 


times  7  cents  in  273^.     Therefore,  at 
balls  of  twine  for  273^. 


a  ball,  we  can  get 


DIVISION.  75 

36.  How  many  pine-apples  can  be  bought  for  $27.78,  or  2778^, 
at  6^  apiece  ? 

Make  27  dots,  and  separate  them  into  groups  of  6  each  ;  thus, 

£»».*«*)  ( )  ( )  ( »)  •  •  • 

Note. — With  this  single  line  of  counters  before  him,  the  pupil  should  now  be 
able  to  answer  the  following  questions : 

1.  How  many  such  lines  would  represent  the  number  of  cents 
expressed  by  the  first  two  figures  of  2778^  ? 

2.  How  many  dots  would  there  be  in  10  such  lines  ?  In  100  ? 
How  many  groups  of  6  in  10  lines  ?  In  100  ?  How  many  over 
in  10  lines?  In  100?  The  300  dots  over  and  78  dots  would 
make  how  many  dots,  etc.  ? 

(From  this  point  the  process  is  exactly  the  same  as  in  the  preceding  Ex.  The 
pupil  may  complete  the  work,  beginning  with  the  making  of  37  dots,  if  necessary,  etc.) 

66.  The  Arithmetical  Solution. — 6  is  not  contained  in  2778 
any  thousands  of  times,  for  even  one  thousand  times  6  would  be 
6000,  but  it  is  contained  some  hundreds  of  times,  for  100  times  6 
is  only  600.     The  question  is,  how  many  hundreds  of  times  ? 

In  a  line  of  27  dots  we  made  4  groups  of  6  dots  each,  in  ten 
lines  (270  dots)  there  would  be  40  groups,  and  in 
100  lines,  400  groups.     Hence,  in  the  arithmetical        6)2778 
process  we  write  4  in  hundreds'  place.     It  shows  4.. 

how  many  hundred  groups  of  6  can  be  made  of 
27  hundred  dots.     There  are  3  hundred  remainiug.     These,  with 
the  78,  make  378.     How  many  groups  can  be  made  of  378  dots  ? 

In  a  line  of  37  dots  we  could  make  6  groups,  with  one  dot 
over,  and  in  ten  lines  we  could  make  60  groups, 
with  10  over.     We  write  the  6  in  tens'  place.     It        6)2778 
shows  how  many  times  10  groups  of  6  each  can  46. 

be  made  of  37  tens. 

The  ten  over  and  the  8  make  18,  in  which  6  is 
contained  3  times.     Hence,  6  is  contained  463         6)2778 
times  in  2778,  ""463 


76 


STANDARD  ARITHMETIC. 


SLATE     EXERCISES 


Find 
37.  4  in 
296 
348 
264 
192 
356 

How 
4  in 

42.  3124 

43.  4796 

44.  5340 

45.  9516 

46.  6779 

47.  8040 

48.  1596 

49.  7120 


how  many  times 
38.  6  in 
258 
462 
354 
408 
516 

many  times 
6  in 

50.  3510 

51.  2578 

52.  1602 

53.  5958 

54.  1086 

55.  2802 

56.  4556 

57.  3062 


39.  7  in 
656 

483 
598 
651 
619 

7  in 

58.  7056 

59.  4606 

60.  6110 

61.  2072 

62.  5408 

63.  4260 

64.  1645 

65.  1113 


40.  8  in 

784 
608 
527 
770 
664' 

8  in 

66.  1707 

67.  9608 

68.  7527 

69.  5768 

70.  3664 

71.  6856 

72.  2392 

73.  2216 


41.  9  in 

399 

432 
873  . 
778 
617 

9  in 

74.  5373 

75.  2961 

76.  4212 

77.  8136 

78.  7008 

79.  4905 

80.  2220 

81.  2052 


Definitions. 

67.  Division  is  a  process  of  finding  how  many  times  one 
number  is  contained  in  another,  or  of  finding  one  factor  of  a 
number  when  the  other  factor  is  known. 

Division  has  two  uses  : 

1.  To  find  how  many  equal  parts  there  are  in  a  number  when 
we  know  what  one  part  is. 

2.  To  find  one  of  the  equal  parts  of  a  number  when  we  know 
how  many  parts  there  are. 

68.  Terms  used. — The  Dividend  is  the  number  to  be  divided. 

69.  The  Divisor  is  the  number  by  which  we  divide. 

70.  The  Quotient  is  the  number  of  times  the  divisor  is  con- 
tained in  the  dividend. 

71.  A  part  of  the  dividend  remaining  undivided  is  called  the 
Remainder.  ^v 


DIVISION.  77 

72.  Signs. — There  are  three  signs  used  to  indicate  division  : 

1.  The  divisor  is  placed  on  the  right  of  the  dividend  with  the 
sign  -f-  between  them  ;  84  -r-  7  is  read  84  divided  by  7 ;  or, 

2.  The  divisor  is  placed  on  the  left  of  the  dividend,  with  a 
curved  line  between  them  ;  thus,  7)84  is  read  84  divided  by  7  ;  or, 

3.  The  divisor  is  written  beneath  the  dividend  with  a  hori- 
zontal line  between  them  ;  thus,  -^  is  read  84  divided  by  7. 

Note  1. — Multiplication  is  taking  one  number  as  many  times  as  there  are  units 
in  another.  Division  is  finding  how  many  times  one  number  is  contained  in  another. 
Hence  division  is  the  reverse  of  multiplication. 

Note  2. — One  number  can  be  taken  from  another  as  many  times  as  it  is  con- 
tained in  it,  hence  by  division  we  find  how  many  times  one  number  can  be  sub- 
tracted from  another. 

Division  of  Numbers  into  Parts, 

73.  If  a  single  thing  or  a  number  of  things  is  divided  into 
two  equal  parts  or  lots,  the  parts  are  called  halves.  Thus,  if  a 
boy  shares  an  apple  equally  with  a  school-mate,  he  cuts  it  into 
two  equal  parts  :  each  part  is  a  half  of  the  apple  ;  or,  if  he  shares 
a  number  of  chestnuts  equally  with  another,  he  divides  the  chest- 
nuts into  two  equal  lots  :  each  lot  is  a  half  of  the  whole  number. 

If  a  single  thing  or  number  of  things  is  divided  into  three 
equal  parts,  the  parts  are  thirds;  if  into  four  equal  parts,  the 
parts  are  fourths,  etc.     Such  equal  parts  are  called  fractions. 

74-.  One  half  is  written  in  figures  thus  :  %  ;  the  figure  above 
the  line  represents  one,  the  figure  below  represents  the  word  half. 

One  third  is  expressed  by  1  above  the  line  and  3  below  it, 
thus,  y3 ;  two  thirds  is  written  %  ;  the  1  and  the  2  represent  the 
number  of  thirds,  the  3  stands  for  the  word  thirds. 


ORAL    AND     ILLUSTRATIVE     EXERCISES. 

l.  A  mother  has  a  basket  of  pears  which  she  wishes  to  divide 
equally  among  three  children  :  what  part  of  the  whole  number 
will  she  give  to  each  one  ?    What  part  would  be  given  to  each  if 


78 


STANDARD  ARITHMETIC. 


they  were  equally  divided  between  2  children  ?  Among  5  children? 
4  children  ? 

2.  If  a  father  divides  8  one-cent  pieces  among  4  children,  what 
part  of  the  number  does  each  one  receive  ?    How  many  is  that  ? 

3.  If  he  divides  18^  among  6  children,  how  many  does  each 
one  receive  ?    What  part  of  the  whole  number  is  that  ? 

4.  Illustrate  with  strips  of  paper,  apples,  or  other  objects,  what 
is  meant  by  %,  y3,  %,  y4,  %,  3/4,  of  anything  or  number. 

5.  Show  that  2/2, 3/3, 4/4,  of  anything  are  equal  to  the  whole  of  it. 

6.  Illustrate  with  objects  what  is  meant  by  3/4  of  24  ;  also, 


four  sixths, 
three  sixths, 
one  sixth, 
two  sixths, 
six  sixths, 
three  tenths, 
five  sixths, 


Vs  of  32, 
V.  of  27, 
V,  of  54, 

M>V/£e  in  figures. 

one  third, 


%  of  28, 
%  of  56, 
y4  of  36, 


three  thirds, 
two  tenths, 
six  sevenths, 
two  sevenths, 
four  tenths, 
six  fourths, 


three  sevenths, 
five  sevenths, 
one  seventh, 
seven  sevenths, 
six  sevenths, 
four  eighths, 
seven  tenths, 


%  of  24, 
%  of  72, 
%  of  50. 


five  eighths, 
eight  ninths, 
seven  eighths, 
five  ninths, 
nine  tenths, 
three  ninths, 
seven  fifths, 


The  Two  Uses  of  Division  Compared. 

75.  The  two  uses  of  division  which  are  represented  in  the 
following  problems  are  often  confounded.  The  figures  employed 
in  the  arithmetical  solutions,  and  the  digits  in  the  answers,  are 
exactly  the  same  for  both,  yet  the  answers  are  really  different, 
and  the  explanations  of  the  process  by  which  they  are  obtained 
should  vary  accordingly. 

l.   How  many  times  is  5  2.  One  fifth  of  45  is  how 

contained  in  45  ?  many  ? 

The  arithmetical  solution  to  both  is  the  same,  thus 
45  -i-  5  =  9. 


DIVISION.  79 

The  Solution  with  Counters. — That  the  pupil  may  clearly  understand  the  essen- 
tial  differences  between  the  two  problems,  and  yet  why  their  modes  of  solution 
should  be  the  same,  let  him  solve  them  with  counters,  thus : 

1.  To  find  how  many  times  5  are  2.  To  find  J/5  of  45  he  does  pre- 

contained  in  45,  he  puts  down  45  cisely  the  same  as  before,  and  then 

counters,  arranging  them  in  groups  takes  one  counter  from  each  group, 

of  5  as  he  does  so.  thus  getting  l/j  of  all  the  groups, 

or  V*  of  45. 


The  answer  to  this  is  9  groups  •  •  • 

of  5,  or  9  Jives.  The  answer  to  this  is  9  units. 

In  this  way  it  is  shown  that  there  are  as  many  units  in  1/5  of 
45  as  5  is  contained  times  in  J+5. 

Show  with  counters  how  to  find 

1.  y3  of  18  4.  %  of  36  7.  %  of  18  10.  %  of  54 

2.  y4  of  28         5.  %  of  42  8.  Vt  of  49  11.  %  of  72 

3.  %  of  30  6.  %  of  55  9.  %  of  40  12.  %  of  54 


OBJECTIVE     EXERCISES. 

Work  out  the  following  problems  with  the  aid  of  counters, 
without  using  your  knowledge  of  the  multiplication  table,  and 
in  each  case  state  whether  it  is  your  object  to  find  how  many 
there  are  in  a  lot ;  or,  how  many  lots  there  are. 

1.  Six  gentlemen  on  a  fishing  excursion  catch  48  fish,  and 
divide  them  equally  among  themselves,  how  many  does  each  one 
receive  ?  How  many  poor  families  would  they  supply  if  6  fish 
were  sent  to  a  family  ? 

2.  A  dairyman  has  96  pounds  of  butter  to  be  sent  to  market, 
how  many  jars  will  he  need  if  he  puts  8  lbs.  in  a  jar  ?  How 
many  pounds  must  he  pack  in  a  jar  if  he  has  but  8  jars  ? 

3.  A  lady  pays  $42  for  14  yards  of  silk,  how  much  does  she 
pay  a  yard  ?  If  instead  of  the  silk  she  buy  velvet  at  $6  a  yard, 
how  many  yards  would  she  get  for  the  same  money  ? 


80 


STANDARD  ARITHMETIC. 


4.  A  teamster  hauls  9  barrels  of  coal  oil  at  a  load;  how  many 
loads  does  he  make  of  126  barrels  ?  He  puts  them  into  3  freight 
cars;  how  many  in  a  car  ? 


ORAL    AND     SLATE     EXERCISES, 

l.  Five  ninths  of  54  are  how  many  ? 

Analysis. — x 

(9  of  54 

is  6 ;  and  5/9  of  54  are  5  times  6  s= 

30. 

How  many 
2.  3/s  of  72 
4/9  of  36 
Vs  of  48 

are 
3. 

%  of  54              4.  %  of  32 

%  of  42                  %  of  27 
%  of  45                   %  of  32 

5.  Vio  Of"  90 

%  of  28 

Vo  of  81 

0.  %  of  72 
5/9  of  63 
V8  of  64 

7. 

*/t  of  63               8.  %  of  35 
%'  of  56                   %  of  48 
3/4  of  32                   Vs  of  20 

9.  %  of  49 

%  of  72 
Vis  of  84 

How  many 
10.  2/5  of  725 

are 

15.  %  of  8008 

20. 

V5  of  18365 

11.  2/s  of  891 

16.  %  of  4392 

21. 

Vs  of  93208 

12.  %  of  582 

17.  "As  of  2196 

22. 

5/7  of  98098 

13.  *// of  861 

18.  7T  of  4011 

23. 

%  of  35172 

14.  3/s  of  520 

19.  9/10  of  7680 

24. 

%  of  31738 

Remainders,  and  how  to  Treat  them. 

76.  l.  To  divide  seven  apples  equally  between  two  persons, 
we  would  divide  6,  the  greatest  number  of  them  that  can  be  so 
divided  without  cutting  any,  and  then  we  would  cut  the  remain- 
ing apple  into  two  equal  parts,  and  give  one  part  to  each  person. 

In   like  manner  a   half  of    7  is  obtained  arith- 
metically  by   first    finding    how   many   there   are   in        2)7 
a  half  of  6,  and  adding  thereto  one  half  of  the  re-  3  7s 

mainder. 

2.  If  11  apples  were  to  be  divided  equally  among  three  chil- 
dren, we  would  divide  9,  the  greatest  number  of  them  that  can 
be  so  divided  without  cutting,  into  3  equal  lots ;  then  cutting 


DIVISION,  81 

each  of  the  two  remaining  apples  into  thirds,  we  would  put  two 
of  the  parts  with  each  lot. 

In  like  manner  the  */3  of  11  is  found  arithmetically      3)11 
by  finding  first  the  %  of  the  greatest  number  in  11  3% 

that  can  be  divided  by  3  without  a  remainder,  and 
then  adding  thereto  */3  of  the  remainder. 


ILLUSTRATIVE    EXERCISES. 

\ 

Show  by  the  division  of  two  strips  of  paper  of  the  same  length 
and  breadth,  or  by  two  lines  drawn  side  by  side  and  of  the  same 
length,  that  2/3  of  1  are  equal  to  the  y3  of  2. 

1.  Divide  7  sticks  of  candy  equally  among  5  children.  Illus- 
trate the  actual  division  by  lines  upon  the  slate  ;  also  perform  the 
example  arithmetically. 

2.  Divide  9  pencils  among  4  boys  ;  7  yards  of  ribbon  among  5 
girls.      (Illustrate  as  above.) 

3.  Divide  11  dollars  among  4  persons ;  13  dollars  among  3 
persons ;  17  dollars  among  5  persons.     (Illustrate  with  counters.) 

4.  Divide  7  feet  into  3,  9  inches  into  4,  13  yards  into  6, 
19  feet  into  7  equal  parts. 

5.  Divide  16  min.  into  5,  21  h.  into  6,  39  d.  into  9  equal  parts. 


SLATE     EXERCISES. 

These   examples   may  be  read  thus  :    find  y3  of  235  ;    */4  of 
457,  etc.,  etc.,  or  how  many  times  3  in  235. 

6.  235-1-3=  14.  4934-3=  22.  5674-8=  30.  4314-8= 

7.  4574-4=  15.  7434-6=  23.  3854-6  =  31.  5074-8= 

8.  3684-5=  16.  6214-7=  24.  7454-6=  32.  620-4-7= 

9.  2794-6  =  17.  349-4-3=  25.  946-4-7=  33.  4704-7= 

10.  4634-7=  18.  5734-4=  26.  8534-7=  34.  5834-6= 

11.  7494-8=  19.  8314-4=  27.  3454-8=  35.  4974-6  = 

12.  8354-9=  20.  7564-5=  28.  7004-9=  36.  3494-5= 

13.  4534-8=  21.  4924-5=  29.  6004-9=  37.  4634-4= 


82                              STANDARD  ARITHMETIC. 

38.  4321-5-3  =  48.  4567-5-5  =  58.  76543-5-4=  68.  44312-4-7: 

39.  5678-4-4=  49.  8765-5-4=  59.  49021-4-5=  69.  57368-5-8: 

40.  4566-5-5  =  50.  9463-4-5=  60.  74935-5-6  =  70.  49564-5-9: 

41.  8321-5-6=  51.  4407-4-6=  61.  68427-5-7=  71.  87310-5-8= 

42.  4720-5-7=  52.  8371-4-7=  62.  54379-5-8=  72.  40302-4-7  = 

43.  5008-4-8=  53.  9462-5-9  =  63.  48628-5-9=  73.  50000-4-4: 

44.  6000-5-9=  54.  4587-5-8=  64.  34567-5-8=  74.  46738-5-6  = 

45.  4725-5-8=  55.  5349-5-9=  65.  50021-4-7=  75.  27493-5-3  = 

46.  9613-5-7=  56.  4623-5-8=  66.  38745-5-6=  76.  56843-5-5  = 

47.  7895-4-6=  57.  7000-5-7=  67.  74960-5-5=  77.  74219-4-8= 


Applications. — l.  How  many  pounds  of  beef  can  be  bought  for 
1854^,  at  9^  a  pound  ? 

2.  There  are  12  boys  on  6  sleds;  what  part  of  the  boys  to  each 
sled  ?  A  regiment  of  531  men  is  transported  in  9  cars ;  how 
many  men  to  each  car  ?     What  part  of  the  regiment  ? 

3.  A  man  divides  $58424  among  his  8  children ;  how  much 
does  each  one  get  ?     What  part  of  the  whole  is  that  ? 

4.  A  carpenter  cuts  a  strip  of  molding  192  inches  long  into 
8  equal  pieces.  How  long  is  each  piece  ?  Suppose  he  cuts  it 
into  4  equal  pieces;  how  long  will  each  be  ? 

5.  There  are  5280  feet  in  a  mile,  how  many  in  y4  of  a  mile  ? 

6.  If  a  boy's  hoop  measures  just  8  feet  in  circumference 
(around  it),  how  many  times  will  it  revolve  (turn  round)  in  a 
half  mile  ? 

7.  If  I  pay  150  for  3  lead-pencils,  what  part  of  the  money  do 
I  pay  for  one  ?  If  I  pay  $15759  for  9  lots,  how  much  do  they 
cost  me  apiece  ?    What  part  of  the  whole  sum  does  each  lot  cost  ? 

8.  If  a  ship  sails  1918  miles  in  two  weeks  at  a  uniform  rate 
of  speed,  what  part  of  the  distance  does  she  sail  in  one  week  ? 
How  many  miles  ?  What  part  of  the  whole  distance  does  she 
sail  in  one  day  ?  How  many  miles  ?  What  part  of  the  whole 
distance  in  two  days  ?    How  many  miles  ? 


DIVISION.  83 

Long  Division. 

77.  l.  How  many  times  is  37  contained  in  9386  ? 

Here  we  have  a  divisor  which  does  not  occur  as  a  factor  in  the 
multiplication  table,  hence  we  construct  a  table  specially  for  it. 
Having  done  this,  we  proceed  with  the  division  exactly  as  we 
do  with  divisors  less  than  10,  except,  1st,  that  we  write  down  the 
products  and  remainders  because  too  large  to  carry  in  the  mind  ; 
and  2d,  that  we  place  the  quotient  over  the  dividend  that  it  may 
be  out  of  the  way  of  the  written  work  which  is  to  follow. 
We  first  find  in  this  table 

Table  of  Multiples  of  37.    that  2   times  37  =  74  J   hence       Process  of  Division. 

1X37=  37        we  know  that  37  is  contained  25325/37 

2  X  37=  74        two  times  in  93,  and  therefore        37)9386 
3X37=111        2  hundred  times  in  93  hurt-  Uoo 

4x37=148        dred.     We  place  the  2  in  the  1986 

5x37=185        order  of  hundreds  (over  hun-  1850 

6X37=222        dreds'  place  in  the  dividend),  136 

7X37=259        and  subtract  the  74  hundred  111 

8x37=296        from  the  93  hundred,  and  ob-  25  Rem. 

9x37=333        tain  the  remainder,  19  hun- 
dred.    Now,  instead  of  carrying  this  to  the  8 

mentally,  we  annex  the  8  to  the  19,  and  thus  obtain  198  tens  for 

the  second  partial  dividend. 

Again,  by  referring  to  our  table,  we  find  that  37  is  contained 
in  198  (tens)  5  (tens)  times,  this  we  write  in  the  order  of  tens, 
and,  subtracting  185  from  198,  we  get  13  (tens)  for  a  remainder. 
To  this  we  bring  down  the  6  units  and  proceed  as  before. 

Thus  we  find  that  in  9386,  37  is  contained  253  times,  with  a 
remainder  of  25.  This  can  be  proved,  for  253  times  37  are  9361, 
and  the  remainder,  25,  being  added  to  9361,  the  sum  is  9386. 

Note. — The  terms  of  the  several  partial  dividends  that  are  at  the  right  of  the 
first  figure  brought  down,  and  the  ciphers  annexed  to  the  several  products,  may  be 
omitted  in  the  work,  since  they  have  no  effect  in  the  result.  In  the  process  above, 
the  figures  that  may  be  omitted  are  printed  in  italics. 


EXERCISES. 

8.  2035 

11.  5817 

9.  8980 

12.  3542 

10.  6050 

13.  9273 

84  STANDARD  ARITHMETIC. 

SLATE 

Find  how  many  times  37  in 

2.  9860  5.  8376 

3.  7935  6.  9098 

4.  5047  7.  5672 

Note  1. — In  the  following  exercises,  14-40,  construct  a  table  of  multiples  for 
each  divisor.  These  exercises  can  be  carried  to  any  desirable  extent.  The  divisors 
remaining  the  same,  the  same  table  of  multiples  will  suffice  for  thousands  of  ex- 
amples. It  will  be  well  to  practice  the  pupils  in  this  way  till  they  are  thoroughly 
familiar  with  the  process  of  long  division.  They  will  then  find  little  difficulty  in 
obtaining  quotient  figures  without  the  aid  of  tables. 

Note  2. — The  table  of  multiples  may  be  formed  by  adding  the  divisor  to  itself 
for  the  second  multiple,  next  by  adding  the  divisor  to  this  sum,  and  so  on,  till  the 
tenth  multiple  is  reached.  If  this  be  the  same  as  the  divisor,  with  a  cipher  annexed, 
the  result  is  a  good  test  of  the  accuracy  of  the  table. 

Find  how  many  times 


54  in 

49  in 

64  in 

83  in 

98  in 

14.  872 

19.    658 

24.  7856 

29.  87506 

34.  296725 

15.  953 

20.  9030 

25.  4785 

30.  84378 

35.  875682 

16.  428 

21.     720 

26.  9378 

31.  59643 

36.  937865 

17.  397 

22.     692 

27.  2704 

32.  23232 

37.  468728 

18.  685 

23.  5377 

28.  1921 

83.  12345 

38.  321485 

39-40. 

How  many  1 

times  98  in  576432? 

1694761?  In  31 

18674930006? 

Definitions. 

78.  When  the  steps  of  the  solution  are  all  written,  as  in  the 
preceding  examples,  the  process  is  called  Long  Division. 

79.  Any  part  of  a  dividend  used  to  obtain  a  quotient  figure 
is  called  a  Partial  Dividend.  (It  is  only  a  part  of  the  entire 
dividend.     See  also  partial  product.) 

80.  The  use  of  the  multiple  tables  is  convenient  when  we 
have  to  employ  the  same  divisor  many  times  successively,  as  in 
the  foregoing  exercises,  but  it  would  involve  a  great  deal  of  un- 
necessary labor  to  construct  one  for  every  divisor  we  may  happen 


DIVISION. 


85 


to  have.  Hence  it  is  desirable  to  learn  how  to  obtain  a  quotient 
figure  without  the  aid  of  such  a  table.  In  doing  this  the  learner 
should  ask  himself,  "What  must  I  multiply  the  divisor  by,  so  as 
to  obtain  a  product  not  greater  than  the  partial  dividend,  and  not 
so  small  that  the  remainder  will  be  equal  to  or  greater  than  the 
divisor  ?" 

Note. — If  the  product  is  greater  than  the  partial  dividend,  the  term  proposed 
for  the  quotient  is  too  great,  and  if  the  remainder  is  equal  to  or  greater  than  the 
divisor,  the  proposed  term  is  too  small. 


SLATE 

EXERCISES. 

1-5. 

6-10. 

11-15. 

16-20. 

234-^-11  = 

5364-31  = 

743-4-62  = 

3456+51; 

548+11= 

6354-41  = 

6344-72= 

2345+51: 

754+21  = 

8744-41  = 

5494-82  = 

3856+51: 

638-4-21  = 

504-j-52= 

6384-53= 

7432^61: 

4974-31  = 

970-4-52= 

5434-95  = 

1579+61: 

21-25. 

26-30. 

31-35. 

36-40. 

3842-4-71  = 

34614-82  = 

234-r-19= 

6844-69: 

65484-71  = 

71114-73  = 

7654-24= 

943+13: 

74324-81  = 

90004-64= 

8014-35  = 

976+25  = 

9465+81  = 

40504-55  = 

7434-46= 

564+36: 

4567-^91  = 

60314-46= 

2574-58= 

310+47= 

41-45. 

46-50. 

51-55. 

56-60. 

2404-58= 

36544-57= 

35794-54= 

64924-88: 

5894-69= 

78904-65= 

13574-29= 

7483+73  = 

4324-88= 

23454-78= 

46824-37= 

6294+97= 

3454-77= 

79374-47= 

70384-76  = 

73854-68= 

6784-59= 

24684-38= 

49254-89= 

4291+51  = 

61-65. 

66-70. 

71-75. 

76-80. 

4064-23  = 

5004-74= 

4000-4-32= 

4000^87= 

7094-34= 

400-r-83  = 

2000+43= 

3000+96= 

3054-54= 

3004-92= 

6000+54= 

40004-65= 

407+56= 

5004-94= 

3000+65= 

7000-j_44= 

8084-65= 

800-4-52= 

6000+76= 

9000+33= 

80 


STANDARD  ARITHMETIC. 


Find  how  many  times 

81.  326  in  1630 

82.  251  in  2362 

83.  347  in  1829 

84.  628  in  2654 

85.  592  in  1867 


96.  428  in  12415 

97.  326  in  24081 

98.  234  in  .13462 


86.  489  in  4375 

87.  981  in  982 

88.  873  in  ^756 

89.  784  in  7830 

90.  892  in  8000 


99.  435  in  15781 

100.  1723  in  344680 

101.  2938  in  357264 


91.  384  in  684 

92.  721  in  1223 

93.  876  in  1676 

94.  988  in  9875 

95.  876  in  8759 

102.  9321  in  993280 

103.  8746  in  785463 

104.  5932  in  593175 


General  Rule  for  Division. 

81.  Utile.— 1.  Write  the  divisor  at  the  left  of  tho  dividend  with 
a  right  curve  between  them. 

2.  For  the  first  partial  dividend  take  only  as  many  figures  at 
the  left  of  the  given  dividend  as  would,  if  considered  apart  from 
the  rest,  express  a  number  great  enough  to  contain  the  divisor. 

3.  Find  the  greatest  number  by  which  you  can  multiply  the 
divisor  to  make  a  product  not  greater  than  this  partial  dividend, 
and  place  it  in  the  quotient. 

4.  Multiply  the  divisor  by  this  number,  subtract  the  product 
from  the  partial  dividend,  and  to  the  remainder  annex  the  next 
figure  of  the  dividend.  If  the  result  is  equal  to  or  greater  than 
the  divisor,  it  is  the  second  partial  dividend,  but  if  less,  continue 
to  annex  figures  from  the  dividend  in  their  order,  placing  a  cipher 
in  the  quotient  for  each  figure  brought  down,  till  a  partial  divi- 
dend is  formed;  or,  till  all  the  figures  of  the  dividend  have  been 
brought  down. 

5.  Proceed  with  the  second  partial  dividend  as  with  the  first, 
and  so  on,  till  all  the  terms  of  the  dividend  have  been  used.  The 
result  will  be  the  quotient  sought. 

6.  If,  after  the  last  division,  there  be  a  remainder,  place  it  with 
the  divisor  underneath,  at  the  right  hand  of  the  quotient. 

Froof.—  Multiply  the  divisor  by  the  quotient,  and  to  the  pro- 
duct add  the  remainder,  if  any.  The  result  will  be  equal  to  the 
dividend  if  the  work  is  correct. 

Note. — The  learner  should  write  each  term  of  the  quotient  over  the  last  figure 
of  the  dividend  from  which  it  was  obtained.  It  will  save  him  from  some  mistakes 
to  which  he  is  liable. 


DIVISION.  87 

Applications.— l.  Distribute  $13425  equally  among  27  sailors. 
How  much  will  each  one  receive  ? 

2.  If  a  locomotive  can  run  513  miles  in  19  hours,  how  far  can 
it  run  in  one  hour  ?  In  two  hours  ?  In  10  hours  ? 

3.  The  steamer  Suevia  makes  the  trip  from  New  York  to 
Hamburg  in  12  days.  The  distance  is  3408  miles.  How  many 
miles  per  day  does  she  make  ?    How  many  in  6  days  ? 

4.  How  many  pounds  are  there  in  352  ounces  ? 

5.  How  many  days  in  3567  hours  ?     In  as  many  minutes  ? 

6.  How  many  hours  in  4628  minutes  ?    How  many  days  ? 

7.  If  20  horses  eat  1940  bushels  of  Oats  in  a  year,  how  many 
will  one  horse  eat  in  the  same  -time  ? 

8.  If  one  boy  picks  a  barrel  of  apples  in  an  hour,  what  part  of 
the  time  ought  it  to  take  two  boys  to  do  the  same  work  ?  Three 
boys  ?  If  one  man  can  dig  a  ditch  in  54  hours,  how  long  will  it 
take  9  men  to  dig  it  ? 

9.  What  number  multiplied  by  23  will  give  36087  ? 

10.  How  many  times  can  27  be  subtracted  from  62397  ? 

11.  If  the  product  of  two  factors  is  21015,  and  one  factor  is 
45,  what  is  the  other  factor  ? 

12.  When  potatoes  are  75^  a  bushel,  how  many  bushels  can  I 
purchase  for  675^  ? 

13.  A  quire  of  paper  has  24  sheets.  How  many  quires  are 
there  in  5631  sheets  ?    In  1436  sheets  ? 

14.  What  is  the  price  of  a  barrel  of  apples,  if  36  barrels  cost  $90  ? 

15.  At  a  post-office  there  are  812  boxes  in  14  rows,  how  many 
are  there  in  a  row  ? 

16.  If  you  weigh  1476  ounces,  how  many  pounds  do  you  weigh  ? 
How  many  pounds  does  your  sister  weigh,  her  weight  being  133 
ounces  less  than  yours  ? 

17.  Bought  897  acres  of  land  for  $77142.  How  much  did  I 
pay  per  acre  ? 


88  STANDARD  ARITHMETIC. 

When  the  Divisor  has  One  or  more  O's  at  the  Right. 

82.  A  boy  employed  at  a  toy-shop  had  to  put  a  large  number 
of  marbles  into  little  canvas  bags,  which  were  to  be  sold  with  the 
marbles.  He  put  ten  marbles  into  a  bag,  and  when  he  had  thus 
filled  ten  bags,  he  put  them  into  boxes,  and  ten  of  these  boxes 
he  put  into  a  basket  to  be  taken  to  the  store-room.  When  the 
work  was  done  there  were 


gy     §s»-  li».  <&#  Sfc*-      •Vir 


Express  the  number  of  marbles  in  figures. 


ILLUSTRATIVE    AND     SLATE     EXERCISES. 

1.  a.  How  many  baskets  full  in  the  lot  of  marbles  above  repre- 
sented, and  what  would  be  left  if  they  were  taken  away  ?  How 
many  marbles  would  remain  ? 

b.  How  many  thousands  in  4765,  and  how  many  over  ? 

Arithmetical  Process. 
11000)4J765 

4-765  Rem. 

c.  What  do  you  notice  on  comparing  the  figures  of  the  quo- 
tient and  remainder  with  the  figures  in  the  dividend  ? 


2.  a.  How  many  times  2  baskets  full  in  the  lot,  and  how 
many  would  remain  if  two  times  2  baskets  full  were  taken  away  ? 
b.  How  many  times  2000  in  4765,  and  how  many  over  ? 

Arithmetical  Process. 

2000)4765(2         Or,         21000)41765 
4000  2-765 

765 

Note. — The  quotient  figure  is  the  same  as  if  4  were  divided  by  2.  The  figures 
of  the  remainder  are  the  same  as  the  3  right-hand  figures  of  the  dividend,  which 
stand  for  the  boxes,  bags,  and  single  marbles  that  would  be  left  if  2  times  2  baskets 
were  taken  away. 


DIVISION.  89 

3.  a.  How  many  times  3  baskets  full  in  the  lot,  and  what 
would  be  left  if  they  were  taken  away  ? 

b.  How  many  times  3000  in  4765,  and  how  many  over  ? 

Arithmetical  Process. 
3000)4765(1         Or,  3|000)4|765_ 

3000  1-1765  Rem. 

1765  Eem. 

Note. — Here  it  will  be  noticed  that  the  result  is  the  same  as  if  4  were  divided 
by  3  and  the  remainder  prefixed  to  the  figures  cut  off  from  the  dividend. 

4.  a.  How  many  boxes  in  the  lot,  including  those  in  the 
baskets,  and  the  single  boxes  represented  ? 

b.  How  many  times  23  boxes  in  the  lot,  and  what  would  be 
left  if  2  times  23  boxes  were  taken  away  ? 

c.  How  many  times  2300  in  4765,  and  how  many  remaining  ? 

Arithmetical  Process. 

2300)4765(2         Or,         23i00)47|65(2 

4600  46 

165  Rem.  165  Rem. 

Note. — The  result  is  the  same  as  if  47  were  divided  by  23,  and  the  remainder 
prefixed  to  the  figures  cut  off  from  the  dividend. 

Hence,  to  shorten  the  work  of  division  when  the  lower  orders 
of  the  divisor  are  filled  with  ciphers,  we  have  the  following 

83.  Rule. — Cut  off  the  O's  of  the  divisor,  and  as  many  figures 
at  the  right  of  the  dividend.  Divide  as  if  the  parts  left  were  the 
entire  divisor  and  dividend.  The  remainder,  if  any,  and  the  figures 
cut  off  from  the  dividend,  form  the  true  remainder. 


SLATE  EXERCISES. 

1.  5674-40=  7.  4478^-80=  13.    67834-80=  19.    34541-4-80= 

2.  8764-50=  8.  2345-4-60=  14.    45714-70=  20.     26483-4-90= 

3.  3934-60=  9.  67894-70=  15.  783514-20=  21.     987654-80= 

4.  5844-70=  10.  34564-80=  16.  462284-30=  22.  1234564-70= 
6.  7484-80=  11.  74824-90=  17.  571354-40=  23.  7001234-60= 
6.  5094-90=  12.  39254-90=  18.  462874-70=  24.  8456794-50= 


93  STANDARD  ARITHMETIC. 

Shorter  Method  of  Computing  in  Long  Division. 

84.  Many  rapid  accountants  dispense  with  written  products 
in  long  division.  They  form  the  remainder  by  writing  down  at 
once  what  the  several  terms  of  the  product  lack  to  make  up  the 
partial  dividend. 

Example. — l.  How  many  times  956  in  3681  ? 

Explanation. — The  work  at  the  left  exhibits  the  steps  of  the 
3         operation  as  already  learned.      How  the   written 

956)3681         product  may  be  dispensed  with  is  shown  in  the  work  3 

2868         at  tne  "Snt>  f°r  w^ich  the  following  wording  is  a  956)3681 

sufficient  explanation.  01  o 

813                                                                          \  813 
Wording. — 18  and  3  are  21  (carry  2); 

15,  17,  and  1  are  18  (carry  1)  ;  27,  28,  and  8  are  36.     Don't  say 
carry  2,  etc.,  but  do  it. 

The  numbers  in  heavy  italics,  occurring  after  the  word  "and,"  are  written 
while  they  are  being  pronounced. 

2.  How  many  times  is  217  contained  in  7507083  ? 

Explanation.— 217  being  contained  3  times  in  750, 
Written  Work.  we  muitiply,  and  write  down  what  the  product  lacks  to 

34594185/217  make  up  750. 

217)7507083  Wording.— 21  and  9  are  30,  carry  3  ;  3, 


6,  and  9  are  15,  carry  1 ;  6,  7. 


997 

Annexing  the  next  figure  of  the  dividend,  we  have 

2058  997  for  the  second  partial  dividend ;  217  being  contained 

1053  in  997  four  times,  we  set  4  in  the  quotient,  and  proceed 

~Tax  as  before. 

Wording. — 28  and  9  are  37,  carry  3  ;  4, 
7,  and  2  are  9  ;  8  and  1  are  9. 


For  Practice  in  the  Shorter  Method. 

How  many  times 

3.  72  in  856  6.     56  in    934  9.  333  in    6931 

4.  83  in  984  7.  Ill  in  5935  10.  235  in    9871 

5.  87  in  899  8.  222  in  7356  11.  354  in  98763 


DIVISION, 


91 


1.  45600-5-100= 

2.  72400-5-300= 

3.  456004-500  = 

4.  836004-700= 

5.  73500-f-900= 

6.  47400-5-400= 

19.  42764-201  = 

20.  5318-5-102= 

21.  37254-305  = 

22.  4943-5-406  = 

23.  37564-507= 

34.  87654-5-743  = 

35.  94615-5-685  = 

36.  34641-5-567= 

37.  64925-5-784= 

46.  3654701-5-4372= 

47.  2043217-7-6482  = 

48.  4700031-5-6395= 

49.  6127421-5-9362  = 


SLATE     EXERCISES 

7.  43000-5-1000= 

8.  234000-j-4000= 

9.  6450004-6000= 

10.  840000-5-8000  = 

11.  375000-4-3000= 

12.  4687000-5-5000= 

24.  35312-^-342  = 

25.  44325-4-429  = 

26.  73812-5-368  = 

27.  44831-^-493  = 

28.  34052-4-504= 

38.  362874-1926= 

39.  400324-1835= 

40.  506074-1749  = 

41.  483254-1683  = 

50.  5432101-4-7408= 

51.  4382146-5-8432  = 

52.  7040047-5-9069= 

53.  2468301-5-7456= 


13.  991204-  590= 

14.  858004-  780= 

15.  2079004-  630= 

16.  108004-  270= 

17.  2090000-4-3800= 

18.  1617000-5-4900= 

29.  365212-4-7040= 

30.  456721-5-8050= 

31.  8439564-9002= 

32.  4334214-6302= 

33.  3465494-5900= 

42.  346819-5-4297= 

43.  5437264-7453  = 

44.  4925704-6853= 

45.  7492564-9469= 

54.  7654321-5-6435= 

55.  5043062-5-4372= 

56.  3489719-5-9348= 

57.  71543274-8745  = 


Self-Testing  Problems. 

Note. — Divide  each  dividend  by  all  the  divisors.     The  remainder,  if  any,  will, 
in  each  example,  be  divisible  by  9. 


1.  53146827  4-549 

2.  61327548  4-558 

3.  1287613534-567 

4.  123456789-5-576 

5.  9876543214-585 

6.  963187452-5-594 

7.  7123456894-711 

8.  7239186454-729 


9.  7913524684-738 

10.  356912478-5-747 

11.  981762345-5-756 

12.  7654321894-765 

13.  781965423-5-774 

14.  7839934564-783 

15.  7923456814-792 

16.  8297134564-828 


17.  8461235794-  846 

18.  8641235974-  864 

19.  7090054744-  882 

20.  4700495704-  918 

21.  3571146364-  936 

22.  9876543214-  954 

23.  9765483214-  972 

24.  981234567-5-  981 


92  STANDARD  ARITHMETIC. 

Applications. — l.  If  a  clock  ticks  29,347,095  times  in  a  year 
of  365  days,  how  many  times  does  it  tick  in  a  day  ? 

2.  Divide  nine  million  nine  hundred  ninety-eight  thousand  five 
hundred  fifty-seven  by  seven  thousand  eight  hundred  forty-two, 
and  write  out  the  answer  in  words. 

3.  The  Valley  Railroad  is  271  miles  long,  and  cost  $5,272,305. 
What  was  the  cost  per  mile  ? 

4.  If  a  farmer  had  138  acres  in  wheat,  from  which  he  har- 
vested 3692  bushels,  how  many  bushels  did  he  raise  per  acre  ? 

5.  A  milliner  cuts  7  pieces  of  ribbon,  each  10  yards  long,  into 
pieces  each  27  in.  long.  How  many  such  pieces  has  she,  and  how 
many  and  how  long  are  the  waste  pieces  ? 

6.  In  a  week  a  boy  gathers  192  bushels  of  apples  ;  how  many 
bushels  does  he  average  per  day  ? 

7.  A  farmer  raises  1875  bushels  of  wheat,  which  he  exchangee 
for  flour,  at  the  rate  of  5  bushels  of  wheat  for  one  barrel  of  flour. 
How  much  flour  does  he  receive  ? 

8.  Find  how  many  gallons  of  milk  in  8  cans  that  hold  respect- 
ively 92,  102,  170,  89,  97,  125,  106,  and  56  pints  ? 

9.  Suppose  that  two  cans  of  equal  size  together  hold  376  pints; 
how  many  gallons  are  there  in  each  ? 

10.  How  many  poor  families  may  be  supplied  from  37  barrels 
of  flour,  allowing  28  pounds  to  each,  a  barrel  of  flour  weighing 
196  pounds  ? 

11.  If  you  take  86  steps  in  a  minute,  how  many  hours  and 
minutes  will  it  take  you  to  walk  38,270  steps  ? 

12.  A  train  of  28  cars  carries  493,920  pounds  of  flour  in  bar- 
rels, each  barrel  containing  196  pounds.  How  many  barrels  to 
each  car  ? 

13.  In  3  weeks,  a  certain  oil-well  is  said  to  have  produced 
35,000  barrels  of  oil.     How  much  was  that  per  day  ? 

14.  How  many  thousand  make  one  million  ? 


DIVISION.  93 

Original  Problems. 

Note  to  the  Teacher. — Let  the  yard-stick  and  foot-rule  be  as  freely  used  as  the 
circumstances  of  the  school  will  allow.  If  the  foot  or  yard  measure  does  not  "  come 
out  even,"  let  dimensions  be  given  in  inches,  but  let  no  account  be  taken  of  the  frac- 
tions of  an  inch  at  present.  No  pupil  should  be  allowed  to  give  a  problem  which  he 
has  not  solved  himself,  and  the  answer  to  which  he  docs  not  know. 

1.  Suppose  yourself  to  be  building  a  pile  of  cubic  blocks,  each 
measuring  one  inch,  foot,  or  yard  on  each  side,  the  pile  to  be  as 
many  inches,  feet,  or  yards,  on  each  side,  as  you  please,  and  ask 
to  know  how  high  you  can  build  it  with  a  given  number  of  blocks. 

2.  If  you  see  an  oil-cloth  or  carpet  with  square  figures  cover- 
ing a  floor,  measure  one  square,  count  the  number  on  the  side  and 
end  of  the  room,  give  the  facts  to  the  class,  and  ask  them  to  find 
how  long  and  wide  the  floor  is. 

3.  Ask  questions  similar  to  these  :  How  many  states  of  the  size 
of  Delaware  might  be  made  out  of  the  state  of  Georgia  ?  How 
many  cities  of  the  population  of  Albany  (N.  Y.)  might  be  made 
of  the  city  of  New  York  ?  The  members  of  the  class  should  hunt 
up  the  facts  for  themselves. 

4.  Give  the  height  of  one  step  of  a  stair-way,  and  the  distance 
from  the  first  to  the  second  floor,  second  to  third,  etc.,  in  some 
house  just  building,  and  ask  how  many  steps  will  be  needed. 

5.  State  the  cost  of  any  number  of  things,  as  yards  of  cloth, 
horses,  etc.,  etc.,  and  the  price  of  one  to  find  the  number.  State 
the  cost  and  the  number,  and  ask  the  price  of  each. 

6.  Construct  questions  about  changing  hours  to  weeks,  equal 
parts  to  wholes,  etc. 

7.  How  many  days  sail  from  to  for  a  vessel  which 

runs  —  miles  per  day  ?     How  many  hours  run  for  a  railway  train 

from  to  ,  at  —  miles  per  hour  ?      (Find  distances  from  your 

text-book  in  geography,  and  rates  of  sailing  from  your  friends.) 

8.  A  railway  train  goes  from  to in  —  hours.     How 

many  miles  an  hour  ? 

Note. — The  newspapers  often  suggest  interesting  problems. 
5 


94 


STANDARD  ARITHMETIC. 


Principles  of  Division. 

85.  A  convenient  number  of  counters  being  equally  distrib- 
uted among  6  or  8  members  of  a  class,  let  the  following  ques- 
tions be  proposed  : 


1.  If  there  were  twice  as  many 
counters  equally  distributed  among 
the  same  pupils,  how  would  each 
one's  share  be  changed?  If  there 
were  only  half  as  many  counters  ? 

2.  If  the  same  number  of  coun- 
ters had  been  distributed  among 
twice  as  many  members  of  the  class, 
how  would  the  share  of  each  be 
changed  ?    If  among  half  as  many  ? 

3.  If  twice  the  number  of  coun- 
ters had  been  given  to  twice  as  many 
members  of  the  class,  how  would  the 
share  of  each  be  changed  ?  If  one- 
half  as  many  had  been  given  to  half 
as  many  persons? 


1.  How  does  it  affect  the  value 
of  a  quotient  to  multiply  the  divi- 
dend by  2,  by  3,  by  any  number, 
while  the  divisor  remains  unchanged? 
To  divide  the  dividend  by  2,  by  3,  etc. 

2.  How  does  it  affect  the  value 
of  a  quotient  to  multiply  the  divisor 
by  2,  by  3,  by  any  number,  while 
the  dividend  remains  unchanged  ? 
To  divide  the  divisor  by  2,  by  3,  etc. 

3.  How  does  it  affect  the  value 
of  a  quotient  to  multiply  divisor  and 
dividend  by  the  same  number  ?  To 
divide  both  divisor  and  dividend  by 
the  same  number? 


ORAL     EXERCISES 

1.  How  many  13's  in  78  ?  In  5x78?  In  %  of  78  ?  In  9x78? 
In  Vs  of  78  ? 

2.  How  many  times  is  18  contained  in  360  ?  %  of  18  in  360  ? 
%  of  18  in  360  ?  4x18  in  360  ?  %  of  18  in  360  ? 

3.  How  many  times  24  in  96  ?  5x24  in  5x96  ?  %  of  24  in  % 
of  96?  24x24  in  24x96? 

4.  Divide  224  by  28  ;  224  by  2x28  ;  224  by  %  of  28  ;  %  of  224 
by  28 ;  3X224  by  3x28  ;  %  of  224  by  %  of  28  ;  13x224  by  13 
X28.  


SLATE     EXERCISES 


How  many  times 

3119  in  1197696 

4316  in  1031524 
7.  4316  in  1031524^-52 
9.  4316  x  718  in  1031524  x  718 


2.  24x3119  in  1197696 
5.  4316-=-52  in  1031524 
8 
10 


3.  3119  in  1197696-1-24 
6.  4316  in  1031524x52 
4316x52  in  1031524 
4316-J-52  in  1031524-4-52 


DIVISION.  95 

Division  by  Composite  Numbers. 

86.  Let  the  pupil  show,  by  various  examples,  that  division 
by  a  composite  number  (product  of  two  or  more  factors)  may  be  per- 
formed by  dividing  successively  by  its  factors.     Thus,  that 

7,756  63)756(12 

9|108  is  equivalent  to 


12 


126 
126 


Divide  in  both  ways 

1.  18576  by  48  3.  30375  by  81  5.  391272  by  56 

2.  37656  by  72  4.  24678  by  54  6.  629937  by  63 

Why  the  results  of  the  two  methods  should  be  the  same,  and 
how  to  deal  with  remainders  when  they  occur  in  the  division  by 
factors,  is  shown  in  the  solution  of  the  following  example. 

7.  Divide  59  by  42. 

Solution  with  Counters. — In  59  counters  there  are  29  twos  and  1  counter  re- 
maining ;  in  29  twos  there  are  9  sets  of  3  twos  and  2  twos  over ;  in  9  sets  of  3 
twos  each  there  are  1  group  of  7  sets  and  2  sets  of  3  twos  remaining;  all  of  which 
is  shown  as  follows. 

/?/7/7/?/7///7/?A//7/?/7r//?/^/?/?/7/?/7/7r//7/(/?/?/7/?/7/ 


Arithmetical  Process.  Explanation. — As  may  be  seen  in  the  illus 

2 

3 


59  tration,  the  first  divisor  is  2  units  and  the  re- 

«q -t  -j            mainder  is  a  unit;  the  second  remainder  is  2 

—  groups  of  2  each,  and  the  third  is  2  larger  groups 

9 — 2  X  <  =  4 .  .  4           of  3  twos  each.     The  sum  of  these  remainders  is 


1 — 2  X  3  X  2  =  12  17,  the  same  as  that  obtained  by  dividing  59  at 

P        -  j       Tiy  once  by  42.     Thus  we  obtain  the  rule  for  dividing 

any  number  by  the  factors  of  composite  divisors. 

87.  Rule, — 1.  Divide  the  dividend  by  any  factor  of  the  divisor, 
divide  the  quotient  by  another  factor,  and  so  on,  till  an  entire 
set  of  factors  has  been  employed. 

2.  If  remainders  occur,  multiply  each  by  all  the  divisors  preced- 
ing the  one  that  produced  it.  The  sum  of  the  products,  added  to 
the  remainder,  if  any,  resulting  from  the  first  division,  will  be  the 
true  remainder. 


96  STANDARD  ARITHMETIC. 

Self-Testing  Exercises. 

To  be  Constructed  by  the  Teacher  for  Dictation. 

Addition  and  Subtraction.— 1.  Write  any  set  of  numbers,  each 
of  which  shall  be  greater  than  the  preceding,  as,  for  example, 
83,      237,      250,      592,      728,      851,      9872,      18589. 
Subtract  the  first  from  the  second,  the  second  from  the  third,  etc. 
To  the  sum  of  the  remainders  add  the  first  number.     If  the  work 
is  correct,  the  sum  will  equal  the  last  number. 

Multiplication. — 2.  Take  any  set  of  numbers  the  sum  of  which 
is  10,  100,  1000,  etc.,  multiply  each  by  any  given  number,  and 
add  together  the  products.      (See  examples  7-18,  p.  64.) 

Division. — 3.  Take  17  and  19.  Annex  Jf  ciphers  to  each.  Di- 
vide each  number  thus  formed  by  the  sum  of  17  and  19.  If  the 
sum  of  the  quotients,  including  fractions,  be  expressed  by  1,  with 
Jf  ciphers  annexed,  the  work  is  correct. 

4.  Separate  any  number  expressed  by  1  with  1/.  ciphers  annexed 
into  any  two  parts,  each  represented  by  If  figures.  Take  any  two 
convenient  smaller  numbers,  as  29  and  88.  Prefix  the  29  to  either 
of  the  former  numbers,  and  the  88  to  the  other  ;  thus, 

003276  and  886724,  or  88327Q  and  006724. 

Divide  both  numbers  of  either  pair  by  68,  that  is,  the  sum  of  29 
and  88  increased  by  1.     The  test  of  accuracy  is  the  same  as  in  3. 

The  last  method  being  somewhat  complicated,  the  following  additional  example 
is  given.     We  divide  b}'  the  shorter  method  for  the  sake  of  space. 

Explanation.-  401682/10|,  59831 1OT/10a 

ll"Z  £  109)43,78814  109)65^1686 

parts  of  100000,  and  183_  1071 

prefix  43  to  one  and  741  906 

by  the  sum  of  43+  59831 10V109  2  216 

65  +  1-  100000  (The  teacher  adds  the  quotients.)         107 

Note. — In  3  and  4,  other  numbers  may  be  substituted  for  those  in  italics. 


CHAPTER  VI. 

MISCELLANEOUS    EXAMPLES. 
Addition,  Subtraction,  Multiplication,  and    Division. 

1.  George  Washington  was  born  in  1732,  and  was  67  years  old 
when  he  died  ;  what  was  the  year  of  his  death  ?  Abraham  Lincoln 
was  born  77  years  later  than  Washington ;  when  was  he  born  ? 
President  Lincoln  lived  56  years  ;  in  what  year  was  he  killed  ? 

2.  In  what  number  is  244  contained  28  times  ? 

3.  How  many  strokes  does  the  hammer  of  a  clock  make  from  1 
till  12  o'clock,  if  it  strikes  only  the  hours  ?    How  many  in  a  day  ? 

4.  A  man  died  leaving  $5200  to  his  wife  and  three  children. 
The  widow  received  $2500,  and  the  children  shared  the  rest 
equally.     How  much  did  each  one  receive  ? 

5.  A  dealer  proposes  to  ship  100000  eggs  in  boxes  containing 
40  dozen  each.     How  many  boxes  will  he  require  ? 

6.  One  hundred  and  thirty-eight  boxes  of  equal  capacity  con- 
tain 76176  eggs.     How  many  dozen  eggs  in  each  box  ? 

7.  If  I  pay  45^  for  lead-pencils,  at  30  each,  how  many  pencils 
do  I  buy  ?     How  many  if  I  pay  50  each  ? 

8.  A  drover  has  $150.  How  many  cows  can  he  buy  at  $50 
each  ?  $25  each  ?  How  many  could  he  buy  at  $45  each,  and  how 
much  would  he  have  left  ? 

9.  A  son  is  born  when  his  father  is  33  years  old.  When 
the  father  is  36  years  old,  how  much  older  is  he  than  the  son  ? 
How  many  times  as  old  ? 


98  STANDARD  ARITHMETIC. 

10.  Twenty-four  sheets  of  paper  make  a  quire.  How  many- 
quires  in  1824  sheets  ?  In  1/2  as  many  sheets  ?  In  y4  as  many  ? 
In  3  times  as  many  ?    In  5  times  as  many  ? 

11.  How  many  hours  are  there  in  9480  minutes  ?    In  twice  as 

many  minutes  ?      (Always  find  your  answer  in  the  shortest  and  most  con- 
venient way.) 

12.  How  many  days  are  there  in  14088  hours  ?  In  ten  times 
14088  hours  ? 

13.  Out  of  796  logs  3980  planks  were  sawed.  How  many 
planks  were  cut  from  each  log,  supposing  them  to  have  been  of 
equal  size  ? 

14.  Four  boys  agreed  to  sweep  a  school-house  two  weeks  for 
$24,  but  at  the  end  of  the  first  week,  three  of  them  gave  it  up, 
and  left  the  remaining  boy  to  complete  the  work.  How  much 
should  the  last  boy  receive  ?    How  much  each  of  the  others  ? 

15.  The  managers  of  an  orphan  asylum  spend  $239  per  year 
for  each  child.  The  expenses  one  year  were  $7170.  How  many 
orphans  in  the  asylum  that  year  ? 

16.  The  manager  of  a  concert  sold  534  tickets  at  $1  apiece, 
936  tickets  at  $2  apiece,  and  257  at  $3  apiece.  He  gave  out  34 
free  tickets.  The  hall  cost  him  $120  rent,  and  for  gas  and  fuel 
he  paid  $19  extra.  How  much  was  left  after  all  expenses  were 
paid,  including  $2100  for  the  performers  ? 

17.  A  congregation  intends  to  build  a  church,  which  is  to  cost 
$12000.  The  collections  already  made  are  $524,  $726,  $837,  $632, 
$439.     How  much  is  lacking  ? 

18.  Mr.  Brown  earns  $28  while  Mr.  Black  earns  $15.  How 
much  will  Mr.  Brown  earn  while  Mr.  Black  earns  $105  ? 

19.  Express  in  words  the  product  of  the  sum  and  difference 
of  8765  and  5678. 

20.  A  train  of  9  cars  has  in  each  car  63  passengers,  of  whom  4 
are  children.  How  many  passengers  altogether,  how  many  adults, 
and  "how  many  children  ? 


MISCELLANEOUS  EXAMPLES.  99 

21.  January  4th,  paid  into  savings  bank,  $14 ;  February  1st, 
paid  in  $13  ;  February  28th,  drew  out  $11  ;  March  14,  paid  in 
$19  ;  March  31st,  drew  out  $25  ;  April  24th,  paid  in  $17  ;  May 
3d,  paid  in  $9  ;  May  25th,  drew  out  $15  ;  June  1st,  paid  in  $16. 
How  much  had  I  then  in  bank  ? 

22.  A  number  of  boys  in  a  work-shop  earn  $7  each  per  week, 
and  an  equal  number  of  younger  ones  earn  $5  each  per  week. 
How  many  boys  are  there  if  their  wages  amount  to  $132  per  week  ? 

Suggestion. — Suppose  they  work  in  pairs,  an  older  and  a  younger  boy  in  a  pair, 
how  much  would  a  pair  receive  ?     How  many  pairs  ? 

23.  How  many  times  can  a  5  gal.  pail  be  filled  from  a  cask 
containing  150  gal.  ?  How  many  times  from  a  cistern  holding 
12  such  casks  ?     24  casks  ? 

24.  On  Tuesday  the  Opera  was  attended  by  2486  persons ;  on 
Wednesday  by  3574  persons.  How  much  more  money  received 
on  the  second  day  than  on  the  first,  at  $2  per  ticket  ? 

25.  Which  is  the  greater,  and  by  how  much,  one  fifteenth  of 
645,  or  one  sixteenth  of  992  ? 

26.  If  a  man  takes  92  steps  in  a  minute,  how  far  will  he  walk 
in  3  hours  if  he  advances  5  feet  in  taking  2  steps  ?  At  the  same 
rate,  how  far  would  he  travel  in  2  days  of  9  hours  each  ? 

27.  A  railroad  conductor  makes  two  trips  every  day  (except 
Sunday)  from  Philadelphia  to  New  York  and  back.  If  these 
cities  are  90  miles  apart,  what  distance  does  he  travel  in  a  week  ? 
In  a  year,  if  he  has  two  weeks  vacation  ? 

28.  How  many  times  will  a  cart-wheel,  16  feet  in  circumfer- 
ence, revolve  in  going  a  mile  (5280  feet)  ? 

29.  A  drover  paid  $780  for  cows  and  sheep.  Of  this  sum  he 
gave  $360  for  9  cows.  If  a  cow  cost  8  times  as  much  as  a  sheep, 
how  many  sheep  did  he  buy  ? 

30.  How  many  feet  of  telegraph  wire  are  needed  to  connect  two 
stations  with  a  double  line,  the  stations  being  37845  yards  apart  ? 
How  many  poles  would  be  required  if  set  45  yards  apart  ? 


100  STANDARD  ARITHMETIC. 

31.  A  healthy  child's  pulse  beats  78  times  a  minute.  How 
often  does  it  beat  in  an  hour  ?    In  4  hours  ?    In  8  hours  ? 

32.  Find  the  smallest  number  that  must  be  subtracted  from 
9904,  to  leave  a  number  that  can  be  divided  by  173  without  re- 
mainder. 

33.  There  are  35  regiments  in  an  army,  averaging  693  men  in 
each;  how  many  men  in  the  army  ? 

34.  If  a  man  earns  $3  a  day,  how  many  weeks  will  it  take  him, 
working  6  days  a  week,  to  earn  $567  ?  $681  ? 

35.  A  man  bought  a  piece  of  land  for  $1564  ;  he  built  a  house 
on  it  for  $642,  a  barn  for  $273,  and  made  other  improvements 
costing  $148.  He  then  sold  it  for  $3000.  How  much  did  he 
gain  ? 

36.  A  railroad  270  miles  long  has  a  station  every  10  miles;  how 

many  stations  has  it  ?     (There  must  be  a  station  at  both  ends  of  a  road.) 

37.  What  is  the  length  of  a  railroad  that  has  18  stations,  at  an 
average  distance  of  17  miles  apart  ? 

38.  A  family  uses  7^  worth  of  milk  a  day.  What  was  the  cost 
of  the  milk  used  the  last  4  months  of  the  year  ?    (See  27,  p.  49.) 

39.  If  a  man  pays  7^  a  year  for  the  use  of  a  dollar,  how  much 
does  he  pay  for  the  use  of  $5  for  one  year  ?  Of  $10,  $50,  $90, 
$200,  $750  ?     How  much  for  each  for  a  half  year  ? 

40.  If  a  man  pays  6<fi  for  the  use  of  a  dollar  for  one  year,  how 
much  does  he  pay  for  the  use  of  $16  for  one  year  ?  How  much 
for  5,  7,  10  years  ? 

41.  If  a  man  pays  8^  for  the  use  of  one  dollar  for  a  year,  how 
many  dollars  can  he  get  the  use  of,  for  a  year,  for  24^,  32^,  44^, 
56^,  68^,  74^  ? 

42.  Sixteen  messenger  boys  are  employed  to  carry  telegraphic 
dispatches,  and  receive  2$  for  each  dispatch  carried.  How  much 
does  a  boy  earn  in  a  week  who  carries  45  dispatches  per  day  ? 
How  much  do  the  16  boys  earn  in  a  week  if  each  one  averages  38 
dispatches  per  day  ? 


MISCELLANEOUS  EXAMPLES 


101 


43.  The  distance  from  New  York  city  to  the  Cape  of  Good 
Hope  is  6670  miles.  When  a  steamer  from  New  York  is  957  miles 
on  its  way,  and  a  steamer  from  the  Cape  is  829  miles  on  its  way, 
how  far  apart  are  they,  if  both  are  on  the  direct  line  between  the 
ports  ? 

44.  A  farm  has  5  fields,  the  1st  containing  89  acres,  the  2d 
101,  the  3d  174,  the  4th  92,  and  the  5th  72  acres.  If  the  farm 
be  re-divided  into  five  fields  of  equal  size,  how  many  acres  will 
each  contain  ? 

45.  If  two  boys  together  have  9  apples,  and  one  has  3  more 
than  the  other,  how  many  has  each  ?  Two  candidates  together 
receive  3579  votes.  How  many  has  each,  if  one  is  elected  by  a 
majority  of  291  ? 

46.  If  a  square  sheet  of  paper  be  cut,  from  corner  to  corner, 
into  two  equal  parts,  we  shall  have  two 
three-sided  pieces,  called  Triangles,  each  of 
which  is  one  half  of  the  square.  From  this 
hint  can  you  reckon  how  many  square  inches 
there  are  in  a 'triangle  such  as  the  one  at  the 
left,  if  the  base  (the  side  on  which  it  stands) 
measures  8  in.  and  the  height  8  in.  ? 

47.  How  many  square  inches  would  such 
a  triangle  contain  if  the  base  and  height  were  each  87  in.  ? 

48.  If  the  above  figure  represents  a  triangular  piece  of  ground, 
the  base  of  which  is  42  ft.,  and  the  height  the  same,  how  many 
square  feet  does  the  lot  contain  ? 

49.  Now,  suppose  we  had  such  a  triangle  as  the  one  at  the 
right;  how  many  square  inches  would 
it  contain  if  the  base  were  12  inches 
and  the  height  8  inches  ? 

50.  How  many  square  yards  does 
a  triangle  contain,  the  base  being 
672  yards,  and  the  height  84 
yards  ? 


^ 

y\ 

x 

y 

X\ 

1 

102 


STANDARD  ARITHMETIC. 


51.  Suppose  that  you  can  lay  4  and  6  books  along  the  edges 

of  a  table,  as  in  this  engraving;  how 
many  books  can  you  lay  on  the  table 
in  one  layer  covering  the  top  ?  How 
many  times  6  books  ?  How  many 
times  4  books  ? 

52.  How  many  could  you  place 

L~  L11^         on  ^ne  ^a^e  m  ^  layers,  5,  7,  10, 

^^  12,  16,  24  layers  ?. 

53.  If  you  could  lay  9  books  end  to  end  along  the  side,  and 
the  width  of  the  table  were  5  times  the  width  of  the  book,  how 
many  books  could  you  put  on  the  table  in  1  layer,  7,  9,  25  layers  ? 

54.  How  many  books  can  you  place  on  a  table  that  is  twice 
the  length  and  three  times  the  width  of  a  book,  if  you  make  the 
pile  15  books  high  ? 

55.  How  many  books  can  you  put  in  a  pile  9  books  long,  7 
books  wide,  and  31  books  high  ? 

56.  How  many  square  blocks,  6  inches  long  and  wide,  can  be 
piled  on  a  platform  72  inches  long  and  48  inches  wide,  if  the  pile 
is  made  30  blocks  high  ? 


57.  This  is  the  picture  of  a  square  board  divided  by  lines 
into  small  squares,  each  supposed  to  be  1 
inch  long  and  1  inch  wide.  How  many 
inches  long  is  the  board  ?  How  many 
inches  wide  ?  Count.  How  many  squares 
on  the  upper  edge  ?  How  many  rows  of 
squares  ?  How  many  squares  on  the  whole 
board  ? 

58.  How  many  square  yards  in  a  lot  23  yards  long  and  10 
yards  wide  ?    In  one  17  yards  long  and  13  yards  wide  ? 

59.  How  many  square  yards  in  a  lot  20  yards  wide  and  30 
yards  long  ?    In  one  180  feet  long  and  270  feet  wide  ?    (How  many 

yards  in  180  feet  ?  In  270  feet?) 


MISCELLANEOUS  EXAMPLES. 


103 


60.  If  the  blocks  represented  in  the  engraving  are  an  inch 

long,  an  inch  wide,  how  many  inches 
long  and  wide  is  the  table  ?  How 
many  such  blocks  can  you  place  in 
1  layer  if  you  cover  the  top  of  the 
table  ? 
L*^  I     Up^  61.  How  many  m  5,  7,  12,  9,  13, 

4,  15,  6  layers  ? 
! M -—      ♦  LP**  62.  Suppose  the   table   to  be  3 

feet  long  and  3  feet  wide,  how  many 
blocks,  a  foot  long  and  a  foot  wide,  can  you  place  exactly  on  the 
front  and  left  edges  of  the  table  ?  How  many  would  exactly 
cover  the  top  ?  How  many  feet  high  would  you  have  to  make 
the  pile  of  blocks  to  make  it  as  high  as  it  is  long  and  wide  ?  How 
many  blocks,  each  a  foot  high,  would  there  be  in  such  a  pile  ? 

63.  Could  you  tell  without  counting,  how  many  blocks  there 
would  be  in  Vs  of  the  pile  ?  %  ?—  How  many  blocks  would  there  be 
in  y3  of  one  layer?    %  of  one  layer?    1/3  of  2  layers?    2/3  of  2  layers? 

64.  If  with  blocks,  each  1  foot  long,  1  foot  wide,  and  1  foot 
high,  you  make  a  pile  12  feet  long,  10  feet  wide,  and  6  feet  high, 
how  many  blocks  will  there  be  in  one  layer  ?  How  many  in  the 
pile? 

65.  How  many  blocks,  each  an  inch  long,  wide,  and  high,  can 
be  placed  in  a  box  measuring  on  the  inside  12  inches  in  length, 
width,  and  depth  ?  If  it  measures  24 
inches  each  way?  36  inches  each  way? 

66.  How  many  cubic  blocks,  meas- 
uring a  foot  each  way,  can  be  sawed 
out  of  a  block  of  stone  measuring  a 
yard  each  way  ? 

Note. — A  block  measuring  a  foot  long,  a 
foot  wide,  and  a  foot  high,  is  called  a  cubic  foot. 
Cut  a  cubic  inch  out  of  a  piece  of  wood,  or  make 
one  of  clay.  Each  side  of  a  cubic  inch  is  a 
square  inch.     Each  edge  is  an  inch  long. 


/ 

J... 

...;.. 

/ 

-]•• 

- 

! 

-L 

/:.. 

/ 

■f 

...L 

.... 

2 

..;... 

;{•• 

.... 

'/■'" 

- 

.....;. 

_ 

?          /:/yt:t':' // 

P 

A'y 

104  STANDARD  ARITHMETIC. 

67.  How  many  blocks,  measuring  1  inch  each  way,  can  be 
cut  out  of  a  block  of  clay  measuring 
12  inches  each  way  ? 


Jt-Vt       -^ "  P'S         68,  Find  how  man^ cublc  feet  in  a 

Lf  J'-.^   r  it  A.      block  of  marble  0  feet  long,  3  feet  wide, 

ffi^%fc&-  "'     ipF"     and  2  feet  thick. 


69.  A  swallow  flies  on  an  average  2640  yards  per  minute.  How 
many  miles  does  it  fly  in  4  hours  ? 

70.  A  pipe  discharging  9  qt.  of  water  every  minute  fills  a 
reservoir  in  4  hours.     How  many  gallons  does  the  reservoir  hold  ? 

71.  A  milkman  brought  12  cans  of  milk  into  town.  Three  of 
these  contained  8  gallons  each,  5  contained  18  gallons  each,  and 
the  others  together  contained  47  gallons.  How  many  pints  of 
milk  did  he  bring  to  town  ? 

72.  A  gentleman  once  said  :  "  If  I  had  saved  only  5<ft  a  day, 
since  I  was  20  years  old,  my  savings  would  now  amount  to  $730." 
How  many  years  old  was  he  ?     (How  many  cents  in  $730  ?) 

73.  A  mill-wheel  makes  4  rotations  in  a  minute.  How  many 
hours  must  it  revolve  to  make  1800  rotations  ? 

74.  In  the  center  of  a  room,  12  feet  square,  lies  a  rug  6  feet 
long  and  5  feet  wide.  How  many  square  feet  of  the  floor  is  un- 
covered ?     (Draw  a  diagram.) 

75.  A  garden,  in  the  form  of  a  rectangle,  is  72  yards  long  and 
50  yards  wide.  In  it  stands  a  green-house  22  feet  long  and  12 
feet  wide,  and  a  summer-house  10  feet  long  and  10  feet  wide. 
How  much  space  of  the  garden  is  not  under  cover  ? 

76.  Mr.  South  worth  received  15  boxes  of  tea,  the  boxes  and 
the  tea  together  weighing  570  lb.  How  much  did  the  tea  in 
each  box  weigh,  if  the  empty  boxes  weighed  each  3  lb.  ? 

77.  $2000  is  to  be  distributed  among  3  persons.  Mr.  A.  is  to 
receive  1/5  of  it,  Mr.  B.  %  and  Mr.  C.  the  remainder.  How  much 
will  each  one  receive  ?'"'•' 


MISCELLANEOUS  EXAMPLES.  105 

78.  A  certain  book  contains  35220  lines.  How  many  days  will 
it  take  you  to  read  it  through,  at  the  rate  of  587  lines  a  day  ? 

79.  If  I  give  153  barrels  of  flour,  worth  $6  a  barrel,  in  ex- 
change for  54  acres  of  land,  how  much  do  I  pay  per  acre  for  the 
land? 

80.  In  a  certain  year  Mississippi  produced  997,576  bales  of 
cotton  (450  lbs.  to  the  bale).     What  was  its  yalue  at  110  a  pound  ? 


81.  Four  boys  at  a  picnic  make  the  following  inventory  (list) 
of  their  property  : 

William  has  100,  20  marbles,  5  arrows,  6  crackers,  4  apples. 
Monroe     "     180,  16         "         3         "       5         "  2 

Harry       "       60,  50         "ft        "   ...  fc,  V*1  2        " 

James  "  220,  18  l<  7  "2  "  4  "' 
Find  the  average  number  of  cents,  the  average  number  of  mar- 
bles, the  average  number  of  arrows,  etc.,  which  they  have  ;  or,  in 
other  words,  find  how  much  money  each  would  have  if  the  prop- 
erty were  equally  divided  amoug  them.  How  many  marbles, 
arrows,  etc.  ? 

82.  The  same  boys  find  the  weight  of  each  to  be  as  follows  : 
William's,  85  lb.  ;  Monroe's,  78;  Harry's,  86;  James's,  92.  What 
is  their  average  weight  ? 

83.  A  butcher  buys  hogs  weighing  severally  268,  372,  283,  356, 
289,  316,  328  lb.      What  is  their  average  weight  ? 

84.  What  is  the  average  value  of  the  hogs  at  40  per  pound? 
What  is  the  value  of  the  7  hogs  at  the  rate  thus  found  ?  Find 
the  value  of  the  hogs  separately,  and  add  them  together. 

85.  Four  boys  wish  to  find  the  average  age  of  their  fathers. 
William's  father  is  52  ;  Monroe's,  48  ;  Harry's,  45  ;  and  James, 
who  has  made  the  calculation,  says  that  his  father's  age  makes 
the  average  age  of  the  four  just  49  years.  How  old  was  James's 
father  ? 

86.  The  population  of  Cleveland  was  92,829  in  1870  ;  ten  years 
later  it  was  160,146.     What  was  the  average  increase  per  year  ? 


106  STANDARD  ARITHMETIC. 

87.  Bought  5  yd.  of  silk,  at  $2  per  yd.  ;  13  shirts,  at  $2 
apiece  ;  6  pairs  of  socks,  at  $0.50  a  pair ;  and  gave  the  merchant 
a  $50  bill.     What  change  did  I  get  ? 

88.  One  omnibus  contains  23  persons,  another  32,  and  a  third 
26.  If  two  persons  leave  one  of  them,  and  11  are  taken  up  on 
the  way,  how  can  the  party  be  so  divided  that  the  omnibuses 
shall  hold  equal  numbers  ? 


89.  Add  352,  6324,  497,  and  723  ;  subtract  3647  from  the 
sum ;  multiply  the  remainder  by  84,  and  divide  the  product  by 
114.     What  is  the  quotient  ? 

90.  Add  2839,  44051,  6273,  78495,  9617 ;  multiply  the  sum 
by  27,  and  from  the  product  subtract  the  sum  of  the  following 
numbers :  1827,  3929,  8272,  13764. 

91.  How  much  is  %  of  96  ?— %  of  96  ?—1/8  of  96  ?  How  much 
is  %  of  9856  ?— %  of  7656  ?— 8/9  of  3618  ? 

92.  From  1000  subtract  367.  Multiply  582  by  the  subtrahend, 
also  by  the  remainder.  Add  together  the  products.  Can  you  find 
what  the  sum  should  be  without  performing  the  work  in  full  ? 

93.  Divide  the  sum  of  the  products  of  (56  X  37)  and  (44  X  37) 
by  100,  and  tell  why  the  digits  of  the  answer  should  be  the  same 
as  those  of  the  multiplicand. 

94.  The  great  bridge  from  New  York  to  Brooklyn  is  suspended 
from  4  cables,  each  composed  of  6300  wires;  each  3578  feet  long. 
How  many  feet  would  all  these  wires  extend  if  laid  end  to  end  ? 
How  many  yards  ?    How  many  miles  ?    (52S0  feet  =  l  mile.) 

95.  If  laid  end  to  end,  how  many  times  would  the  wires  of 
the  bridge  extend  from  New  York  to  Philadelphia  (90  miles)  ? 
From  New  York  to  Chicago  (980  miles)  ?    From  New  York  to 

San  Francisco  (3400)  ?      (Work  in  miles,  reject  remainders.) 

96.  In  90  years  the  total  population  of  the  United  States  in- 
creased from  3,929,214  to  50,155,783.  What  was  the  average 
increase  per  year  ?    Per  decade  ?    (A  decade  is  10  years.) 


CHAPTER    VII. 

UNITED    STATES    MONEY. 

88.  The  money  of  the  United  States  consists  of  coins  made 
at  the  United  States  mints,  and  government  or  bank  notes  duly 
authorized  by  law. 

Coin  is  the  specie  or  metallic  currency,  and  notes  are  the  paper  currency,  of  the 
country.     (Compare  the  words  current  and  currency.) 

The  pupils  should  be  required  to  write  for  themselves  a  list  of  the  coins  used, 
being  left  to  get  the  information  as  they  can. 

89.  The  following  are  the  denominations  of  money  used  in 
business  and  in  accounts  : 

10  Mills  =  1  Cent,   100  Cents  =  1  Dollar. 

This  table  is  the  correct  business  form,  but  the  decimal  character  of  our  cur- 
rency is  better  shown  in  the  following  table: 

10  Mills  =  1  Cent,  10  Cents  ==  1  Dime,    10  Dimes  =  1  Dollar. 

The  word  mill  is  from  the  Latin  mille,  a  thousand  (1000  m.  =  $1).  Cent  is 
from  the  Latin  centum,  a  hundred  (100^  =  $1).  Dime  is  from  the  old  French 
word  disme,  ten  (10  dimes  =  $1). 

90.  The  sign  for  dollars  is  $  ;  parts  of  a  dollar  are  indicated  by 
a  point  called  a  separatrix.  Both  signs  are  prefixed  to  the  figures 
to  which  they  belong. 

Dollars  are  read  as  one  number,  the  first  two  places  to  the  right 
of  the  separatrix  as  cents,  the  third  as  mills. 

Thus,  $32,875  is  read  32  dollars,  87  cents,  5  mills. 

Note. — For  illustrations  the  hundred*  of  jackstraws  already  used  will  serve  to 
represent  dollars ;  the  tens,  dimes  ;  single  sticks,  cents ;  and  tentlis  of  a  stick,  mills. 
There  is  plenty  of  room  for  devising  other  and  better  illustrations. 


108  STANDARD  ARITHMETIC. 

ORAL    AND     SLATE     EXERCISES. 

1.  Read  $1.00,  $3.05,  $5,  $1.25,  $2.75,  $6.50,  $4.16, 
$15.13,     $16.12,      $71.10,      $35.80,      $70.65,      $40.50,      $38.15, 

$49.35,     $586.50,     $9828.75,     $16782,     $0.25,     $0.05. 

2.  Read  $1,125,  $3,367,  $6,875,  $4,126,  $19,625,  $18,163, 
$3,185,    $72.05,    $1.05,    $1,055,    $1.47,    $0,876,    $2.10,    $342.21. 

3.  Write  in  figures,  and  without  the  aid  of  words,  one  cent, 
two  cents,  etc.,  to  fifty  cents.  (Use  the  sign  $,  thus  $0.01  or  $.01. 
Arrange  in  columns.) 

4.  Write  7  dols.  60  3  m.,  12  dols.  700  8  m.,  88  dols.  600  7  m., 
100  dols.  90  9  m.,  3426  dols.  10  5  m.,  4870  dols.  300  5  m. 

Model.— 7  dols.  60  3  mills  =  $7,063. 

5.  Write  in  words,  326,  $326,  $326.01,  26,  $0.26,  $26.00, 
$2.60,  845,  $8.45,  $84.05,  $0,845,  $16,  160,  $0,169,  $16.05, 
$1,605.      (Write  abstract  numbers  and  sums  of  money  in  separate  columns.) 


6.  How  many  cents  in  $1,  $2,  $3,  $4,  $5,  $9,  $6,  $8,  $7, 
$10,     $11,     $15,     $17,     $25,     $81,     $67? 

7.  How  many  cents  in  $1,  $1.05,  $1.10,  $2,  $2.25,  $3.75, 
$4.85',  $6.10,  $4.17,  $8.25,  $10.20,  $1.99,  $16,' $16.87,  $35.50? 

8.  Write  the  following  as  cents:  $1.75,  $6.25,  $7.35,  $5.45, 
$7.95,    $6.87,    $7.21,    $18.38,    $19.55,    $18.76,    $27.87,    $98.63. 

Model.— $21.83  =  21830.     Ecad  :  21  hundred  83  cents. 

9.  Read  the  following  as  hundreds  of  cents  and  cents,  and  again 

as  dollars  and  Cents  :    16750  (16  hundred  75  cents,  and  16  dollars  75  cents), 

5000,    6820,    9560,    18710,    19560,    45630,    6270,    18560,    39830, 
99870,    86720,    913740. 

10.  Write  the  following,  separating  the  dollars  from  the  cents, 
and  use  the  proper  signs  :  7820,  3960,  9500,  6800,  7210,  3540, 
7250,  8750,  19820,  16780,  25850,  43980,  18750. 

Model.— 876380  =  $876.38.  (The  signs  0  and  m.  must  be  omitted  when  the 
dollar  mark  and  the  separatrix  are  used.) 


UNITED  STATES  MONEY.  109 

11.  How  many  mills  in  10,   20,   30,  4fc   90,  50,  80,   60,   70, 
100  ?    Is  the  mill  coined  ? 

12.  How  many  mills  in  200,  ?O0,  300,  600,  400,  900,  500,  150, 

360,  580,  420,  63^,  510? 

13.  How  many  mills  in  1000,  3000,  6000,  4000,  1200,  1500, 

$1,  $2,  $3,  $16,  $9,  $1.20,  $1.50,  450,  4500,  $4.50,  $9,  $10,  $15, 
$19,  $18.25,  $25.62,  $342.15,  $2100.50  ? 

14.  Express  as  mills  : 
$15 


170 

1630 

$1.75 

200 

2450 

$2.25 

250 

3500 

$3.50 

830 

4250 

$4.05 

Models 

.—160  = 

: 100  m. ; 

$15.12 

$185.16 

$232,132 

$23.19 

$198.21 

$830,012 

$62.20 

$321.15 

$700,019 

$70.00 

$568.00 

$109,346 

$1.35  =  1350  m. ;  $275.18  =  275180  m. 

15.  Read  the  first  three  columns  below  as  cents,  or  as  cents 
and  mills  ;  the  last  three  as  dollars,  cents,  and  mills  : 

10  m.  15  ra.  152  m.  2645  m.  1325  m.  20175  m. 

20  ra.  26  ra.  235  ra.  1875  ra.  2515  m.  38625  ra. 

40  m.  55  m.  428  m.  1625  m.  7628  m.  55825  m. 

50  m.  32  ra-  632  m.  4935  m.  4932  m.  70362  ra. 

16.  Write  the  sums  expressed  above,  using  the  dollar  sign  and 
the  separatrix. 

Model.— 10  m.  -  $0.01.     70362  m.  es  $70,362. 

91.  By  this  time  the  pupil  will  perceive  that  any  sum  of 
United  States  money  may  be  read  in  different  ways.  For  in- 
stance, $1  may  be  read  as  1000  or  as  1000  m. — $16.85  may  be 
read  as  written,  or  as  16850,  or  16850  m. 

92.  The  whole  process  of  changing  one  denomination  in 
United  States  money  into  another  consists  in  annexing  0's,  or  in 
placing  or  removing  the  separatrix.  If  any  denomination  lower 
than  dollars  is  to  be  expressed,  cents,  including  dimes,  must  al- 
ways have  two  places,  even  though  they  be  filled  with  0's.  If  the 
number  of  cents  be  less  than  10,  the  figure  next  to  the  separatrix 
must  be  a  0.     It  is  very  important  to  remember  this. 


HO  STANDARD  ARITHMETIC. 

Addition  and  Subtraction  of  United  States  Money. 
93.  The  following  rule  needs  no  introduction  : 

Mule. — Write  the  numbers  to  be  added  or  subtracted  so  that 
the  separating  points  shall  be  in  a  vertical  line.  Then  proceed 
as  in  addition  or  subtraction  of  simple  numbers. 


SLATE 

EXERCISES. 

1. 

2. 

3. 

4. 

5. 

$  3.75 

$  7.30 

$     3.25 

$  51.32 

$661.50 

4.52 

13.15 

114.29 

46.39 

720.70 

13.30 

2.99 

50.01 

54.91 

43.35 

25.09 

48.75 

36.36 

284.03 

438.97 

Write  in  columns,  and  add : 

6.  $64.22+$1.01+$6.93+$101+$0.28+$57. 05+34. 01. 

7.  $242.32+$4.00+$91.35+$628.21+$5.05+$40+$34.10. 


8. 

9. 

10. 

11. 

12. 

$752.24 

$324.40 

$  13.80 

$13428.79 

$     12.00 

6.23 

276.19 

14.60 

12469.31 

10.00 

15.62 

1328.45 

160.32 

3872.84 

960.05 

103.78 

9327.71 

369.49 

92476.75 

200.91 

923.60 

1000.01 

765.30 

1328.62 

1342.78 

Find  the  sum  of 

13.  $3.421+171.243+162. 103+15. 009+134.24+16.328. 

14.  $20. 21+$342. 109+$34. 497+$603284. 679+$9. 384. 


15. 

16. 

17. 

18. 

$  12.345 

$  10.102 

$793,104 

$     1.000 

67.890 

1.000 

38.796 

342.060 

35.719 

100.309 

596.647 

3.501 

864.246 

540.701 

764.521 

917.624 

395.713 

724.309 

990.901 

249.387 

54.575 

99.830 

881.725 

1800.000 

983.600 

786.250 

9651.062 

71.875 

19.  Add  17  dols.  30^  4  m.,  123  dols.  45^  7  m.,  129  dols.  37# 
m.,  98  dols.  11^  6  m.,  59  dols.  24^  7  m.,  3482  dols.  99^  9  m. 


UNITED  STATES  MONEY.  '   111 

20.  Find  the  sum  of  $5,097  +  $0,083  +  $5,629  +  $15,912  + 
$4,691  +  $59.03  +  16.001  +  $18,875. 

21.  Find  the  sum  of  $0,187  +  $9,987  +  $4.46  +  $0,365  + 
$73.28  +  $83.18  +  $1.  +  $5000  +  $3,057  +  $15.01. 

22.  Add  five  hundred  sixty-nine  dols.  forty-seven  cents,  seven 
thousand  two  hundred  ninety-eight  dols.  sixty-three  cents  five 
mills,  ninety-six  dols.  fifty-three  cents  seven  mills,  one  hundred 
dols.  forty-five  cents  seven  mills,  nine  hundred  dols.  ten  cents, 
one  dol.  fifty  cents. 

Applications. — 23.  A  gentleman  owes  the  following :  To  Mr. 
Brown,  $16.50;  to  Mr.  Jones,  $79.54;  to  Mr.  Thomas,  $82.35;  to 
Mr.  Alfreds,  $46.83.     "What  is  the  sum  of  these  debts  ? 

24.  Mr.  Willey  gave  the  following  sums  to  be  used  for  chari- 
table purposes  :  $3470.25  ;  $6945.75  ;  $1015.00  ;  $5900.25.  How 
much  did  he  give  in  all  ? 

25.  The  property  of  Mr.  Childs  was  valued  as  follows  :  Land, 
$3560  ;  house,  $1834 ;  barn,  $690  ;  horse,  $180  ;  carriage,  $135  ; 
tools,  $382.95  ;  money  in  bank,  $987.     How  much  was  he  worth  ? 

26.  Mr.  Johnson  bought  a  set  of  furniture  for  the  parlor, 
costing  $385.75  ;  one  for  the  sitting-room,  costing  $229.55  ;  one 
for  a  bed-room,  costing  $176.25  ;  one  for  the  dining-room,  costing 
$194.40.     What  was  the  total  cost  of  the  four  sets  ? 

Subtract : 

27.  28.  29.         30.         31. 

$642.67  $648.1G  $3247.67  $162.38  $6235.41 

83.90  59.23  843.98       37.84  1987.93 

32.        33.  34.        35.        36. 

$639.10     $794.13        $9.00      $85.00     $9500.05 
234.78      197.25        2.97        3.78      3428.69 


29. 

$3247.67 

843.98 

34. 

$9.00 

2.97 

39.       40. 

$523.42    $6,932 

94.51     0.478 

37.       38.       39.       40.       41.       42. 

$953.24    $123.45    $523.42    $6,932    $173.00    $11,000 
735.38     67.89     94.51     0.478     56.43     9.345 


35. 

$85.00 
3.78 

41. 

$173.00 
56.43 

112  STANDARD  ARITHMETIC. 

43-50.  Deduct  $87.93  from  each  of  the  following:  $100.00, 
$93.13,    $703.24,    $103.10,    $1834.29,    $793.11,    $600,    $87.94. 
51-55.  Deduct  $934,782  from  the  footings  of  examples  8-12. 
56-59.  Deduct  $379,999  from  the  footings  of  examples  15-18. 

Subtract : 

60.                            61.  62.  63. 

$762,949  $564,120  $24,500  $724,380 

438.758  259.437  19.999  149.871 

64.  Take  four  thousand  six  hundred  forty-five  dollars  twenty- 
eight  cents  three  mills  from  six  thousand  two  hundred  thirty- 
eight  dollars  eleven  cents. 

Applications. — 65.  If  a  person's  property  amounts  to  $7434. 90, 
and  his  debts  to  $1350. 78,  how  much  is  he  worth  ? 

66.  Mr.  White  opened  his  store  with  goods  worth  $6100.50. 
Six  weeks  afterward  he  took  an  inventory  (list)  of  goods  remaining 
on  hand,  and  found  their  value  to  be  $4417.46.  How  much  had 
he  disposed  of  ? 

67.  One  season  Mr.  Hardy,  who  made  a  business  of  dealing  in 
real  estate  (houses  and  lands),  bought : 

a.  A  city  house  and  lot  for  $8765,  and  sold  them  for  $10000  ; 

b.  A  farm  for  $16850,  and  sold  it  for  $17050  ; 

c.  A  business  block  for  $78500,  and  sold  it  for  $82000  ; 

d.  Two  city  lots  for  $30500,  and  sold  one  for  $13250  and  the 
other  for  $18525  ; 

e.  A  city  lot  for  $75000  and  sold  it  for  $83500  ; 

/.  A  theatre  for  $125250,  and  sold  it  for  $121750.  Find  what 
he  gained  or  lost  on  each  transaction.  Did  he  gain  or  lose  by  the 
business  of  the  season,  and  how  much  ? 


68.  How  many  cents  are  there  in  one  half  dollar  ?  In  one 
quarter  of  a  dollar  ?  In  one  fifth  of  a  dollar  ?  In  one  tenth  of  a 
dollar  ?  In  three  quarters  ?  In  four  quarters  ?  In  two  quarters  ? 
In  two  fifths  ?    In  three  tenths  ? 


UNITED  STATES  MONEY.  113 

69.  Write  the  following  sums   as  dollars  and  cents  :   $1%, 
$2%,  $3%,  $5%,   $18%,    $22%,  $28%,   $32%,    $42 %,  '  $48 %. 

Model.— $174  =  $1.25. 

70.  Write  the  number  of  mills  equivalent  to  12  y2*,    182/5*, 
37%*,    75*,  '43%*. 

Model.— 72*=5  m.,  12720=1205  m.  =  125  m. 

71.  Find  the  sum  of  $1%,    $3%,    $5%,    $7%,    $90  %   $10%. 

(Write  $1.75  for  $l3/4,  etc.) 

72.  Add  $7%,    $8%,    $1%0,   $3%0,    $6%,    $30%. 


Applications. — 73.  William  has  $7  %>  in  his  savings  bank,  Ber- 
tha $23/4,  Carl  $1%,  May  $3%,  Emma  $0.50.  How  much  have 
they  in  all  ? 

74.  Mrs.  Duncan  paid  the  butcher  on  Monday  $0.60;  on 
Tuesday,  $0. 78  ;  on  Wednesday,  %,  dol. ;  on  Thursday,  %  dol. ; 
on  Friday,  %  dol.;  and  on  Saturday,  $1.10.     How  much  in  all  ? 

75.  My  new  Eeader  cost  %  dol.,  my  Speller  %  dol.,  my  Arith- 
metic %  dol.,  and  my  new  slate  15*.  How  much  did  I  pay  for 
them  all  ? 

76.  I  bought  a  pair  of  shoes  for  $4%,  a  hat  for  $7%,  and  an 
umbrella  for  $1%,  and  gave  the  merchant  a  $20  bill.  What 
change  did  I  get  ? 

77.  In  paying  an  account  of  $344/5,  I  gave  the  dealer  4  ten- 
dollar  bills.  What  change  was  due  me?  If  I  had  given  him  a 
$50  bill,  what  change  should  I  have  received  ? 

78.  A  boy  had  bought  four  articles  at  the  grocery  worth  $2.25. 
But  not  having  so  much  money,  he  handed  back  one  of  the  arti- 
cles costing  9/io  of  a  dollar.  How  much  did  he  pay  for  the  other 
three  articles  ? 

79.  John  buys  coffee  for  11%,  sugar  for  70*,  cheese  for  % 
dollar.  He  sells  the  grocer  potatoes  for  $1.50.  How  much  does 
John  have  to  pay,  after  deducting  the  price  of  the  potatoes  from 
his  bill  ? 


114  STANDARD  ARITHMETIC. 

Multiplication  of  United  States  Money. 

Illustrative  Example.— l.  If  the  price  of  rye  is  $1,355  per  bu., 
what  will  65  bushels  cost  ? 

Suggestive  Questions. — If  $1,355  were  written  Operation, 

in  column  65  times,  and  an  addition  of  all  made,  *-,   qkk        •                u 

where  would  the  separatrix  fall  ?    How  many  places  * lt 6i)b  PrlCe  Per  DU* 

would  there  be  to  the  right  of  the  point  V     What  65  no.  of  bushels. 

denominations  would  they  represent  ?  6775 

0?*,  How  many  mills  are  there  in  $1,355  ?     In  q-.  qrj 

65  times  $1,355  ?     Keduce  the  result  to  dollars. 

How  many  places  to  the  right  of  the  point  ?    What  $88,075  Cost  of  65  bu. 
denominations  do  they  represent  ? 


94-.  Rule.—  Multiply  as  in  simple  multiplication.  The  product 
will  be  of  the  same  denomination  as  the  lowest  order  of  the  mul- 
tiplicand. If  it  be  in  cents  or  mills  reduce  to  dollars  and  prefix 
the  dollar  mark. 


SLAT  E     EXERCISES. 

2-7.  Multiply  $87.74  by  27.— $324,034  by  56.— $1.95  by  18.— 
$27,341  by  35.— $0,934  by  746.— $0.34  by  61. 

8-22.  Multiply  each  of  the  following  sums  of  money  by  8  ;  by 
37  ;  by  368  :  $0,398  ;  $20.03  ;  $47,731  ;  $621.70  ;  $0,604." 

23-37.  Find  first  13,  then  49,  then  387  times  $34.75; 
$967.03;  $309.08;  $7654.60;  $190.10;  and  subtract  $99,999 
from  each  product. 

38.  What  is  the  product  of  $4.37  by  18x27x15  ? 


Applications. — 39.  How  much  must  the  government  pay  for 
327  horses,  at  $135.50  each  ? 

40,41.  What  will  426  sheep  cost  at  $4.87%  per  head?     At 

$3.62  1/2  ?     (Express  fractions  of  cents  in  mills.) 

42.  What  is  the  amount  of  a  contribution  if  157  persons  con- 
tribute each  $5  y4  ? 

43-45.  What  will  476  lb.  of  tea  cost  at  65^  a  lb.  ?  At  50^  ? 
At%dol.? 


UNITED  STA  TES  MONEY.  115 

46.  A  merchant  bought  3  bales  of  cloth  containing  respectively 
485,  492,  and  497  yards,  at  $1.87  %  a  yard.     Find  the  cost  ? 

47.  The  merchant  just  spoken  of  sold  the  two  bales  first  men- 
tioned at  $2.25,  and  the  last  one  at  $2.75  per  yard.  What  did  he 
receive  for  the  cloth  ? 

48.  A  grocer  buys  38  doz.  loaves  of  bread  on  Monday,  35  doz. 
on  Tuesday,  and  36  doz.  on  each  remaining  day  of  the  week,  at 
60$  per  doz.,  and  sells  at  7^  per  loaf.     How  much  does  he  gain  ? 

49.  A  grocer  bought  9  barrels  of  cider,  each  barrel  containing 
30  gallons,  at  12 1/2(f  per  gallon.     What  did  the  cider  cost  ? 

50.  What  is  the  cost  of  4  barrels  of  sugar,  weighing  495  lb. 
each  at  7  y8#  a  pound  ? 

51.  What  is  the  cost  of  156  acres  of  land  at  $20.50  per  acre  ? 

52.  Mr.  Jones  bought  of  Messrs.  Taylor,  Kilpatrick  and  Co., 
16  yd.  silk,  at  $2.25  a  yd.;  6  yd.  gingham,  at  $0.19  a  yd.;  27 
yd.  linen,  at  $0.37  a  yd.;  39  yd.  muslin,  at  $0.13  a  yd.  What 
was  the  amount -of  the  bill  ? 

53.  Mr.  Taylor  bought  of  Mr.  Watts  the  following  implements  : 
a  spade  for  3/4  dol.,  a  rake  for  %  dol.,  a  hoe  for  %  dol.,  a  shovel 
for  45^',  and  a  lawn-mower  for  14%  dols.     Find  the  amount. 

54.  Mr.  James  went  to  market  with  $5.  He  spent  for  eggs 
45^,  for  butter  98^,  for  fruit  25^,  and  for  flour  $1.20.  How  much 
had  he  left  of  the  $5  ? 

55.  A  laborer  earns  $50  and  spends  $39.75  a  month.  How 
much  will  he  have  saved  at  the  end  of  six  months  ? 

56.  What  are  the  profits  of  a  concert,  if  3427  tickets  are  sold 
at  $1  y2  each,  and  the  expenses  are  $938.40  ? 

57.  Mr.  Mills  sold  837  shade-trees  for  $0.65  each.  How  much 
did  he  receive  for  the  lot  ? 

58.  A  drover  bought  265  head  of  cattle  at  $43.75  a  head,  and 
paid  $8.35  a  head  to  get  them  to  market,  where  he  sold  them  at 
$56.80  a  head.     How  much  did  he  gain  by  the  transaction  ? 


116 


STANDARD  ARITHMETIC. 


Division  of  United  States  Money. 

ILLUSTRATIVE     EXAMPLES. 


1.  If  a  builder  pays  $693.68 
for  lumber  at  %t]^f  per  foot, 
how  many  feet  does  he  buy  ? 

27747  %5  times. 

25  m.)693680  m. 
50 

193" 
175 

186 
175 

118 
100 

180 
175 


2.  If  $693.68  be  equally  di- 
vided among  25  men,  what  will 
be  the  exact  share  of  each  ? 

$27.7475/25 

25)1693.68 
50 

193 
175 

186 
175 


118 

100 


Note  1. — Every  time  2!/2^  can  be 
taken  from  $693.60,  a  foot  of  lumber 
can  be  bought.  Hence,  to  find  the  num- 
ber of  feet,  we  find  how  many  times 
21/2$  is  contained  in  $693.68.  To  do 
this,  we  change  both  the  2  x/2^  an(i  the 
$683.68  to  mills,  and  divide  the  second 
by  the  first. 

A  shorter  way  would  be  to  change 
both  sums  to  half  cents,  and  divide,  but 
the  first  way  is  generally  the  better.  Try 
both  and  see  which  you  like  best. 

Note  3. — Thus  we  find  that  division  of  United  States  money  is,  first,  the  process 
of  finding  how  many  times  one  sum  of  money  is  contained  in  another;  and,  second, 
of  finding  a  required  part  of  a  given  sum.     (See  also  Art.  75,  p.  78.) 

95.  Rule  I. — To  find  how  many  times  one  sum  is  contained  in 
another,  change  both  to  the  lowest  denomination  in  either,  and 
divide  as  in  simple  division.     The  quotient  will  be  in  integers. 

Rule  II  — To  find  a  required  part  of  a  given  sum  of  money, 
divide  the  sum  by  the  number  of  parts,  as  in  simple  division.  The 
place  of  the  separatrix  in  the  quotient  will  be  directly  over  the 
separatrix  of  the  dividend. 


180 
175 

5 

Note  2.— If  $693  be  divided  into  25 
equal  parts,  there  will  be  $27  in  each 
part,  and  $18  undivided.  $18  =  180 
dimes,  180  dimes  4-  6  dimes  =  186 
dimes.  If  186  dimes  be  divided  into  25 
equal  parts,  there  will  be  7  dimes  in 
each,  and  11  dimes  will  remain  undi- 
vided. To  this  we  add  the.  8^,  and  then 
proceed  as  before  till  we  come  to  a  re- 
mainder of  5  mills.  For  the  present,  re- 
mainders should  be  disposed  of  as  direct- 
ed in  Art.  76,  p.  80. 


UNITED  STATES  MONEY,  117 

SLATE     EXERCISES. 

3.  Divide  $100.50  by  5.     Also  by  7,  by  9,  by  65. 

4,  5.  Four  persons  are  to  have  equal  shares  of  $4412.88.  How- 
much  will  each  one  receive  ?  How  much  would  each  receive  if 
there  were  12  persons  ? 

6-11.  Divide  $369,009  by  3,  by  9.— Also  by  8,  by  6,  by  5,  by  15. 

12-16.  Divide  $3759.91  by  19,  by  35,  by  54,  by  67.— Are  you 
here  required  to  find  certain  parts  of  $3759.91;  or,  how  many 
times  that  sum  contains  the  several  divisors  ? 

17.  Which  is  greater,  %  or  %  of  $90.50  ?    How  much  ? 

Find 

18.  y2  of  $     97.78.        22.  Vic  of  $8775.36.        26. 13/16  of  $10391.52. 

19.  y4  of  $  363.68.         23.  %  of  $  436.50.        27.  15/27  of  $  8335.71. 

20.  y5of  $  728.15.         24.  %  of  $  410.41.        28.  %<jf  I  5654.76. 

21.  %  of  $8257.25.         25.  %  of  $9873.56.        29.  18/25  of  $  9864.75. 


Applications. — 30-31.  How  much  sugar  can  be  bought  for  99^ 
at  11^  a  lb.  ?    At  9^  a  lb.  ? 

32-35.  If  oranges  cost  4y>^  apiece,  how  many  can  be  had  for 
$3.60,  for  $3.78,  for  $36.00,  for  $0.90  ? 

36.  A  dozen  chairs  can  be  bought  for  $11.40.  How  much  does 
1  chair  cost  ? 

37.  A  carpenter  has  34  men  at  work  ;  at  the  end  of  the  week, 
17  of  them  receive  $211.65  wages ;  the  other  17  receive  only 
$184.45.  How  much  does  each  one  of  the  two  classes  of  work- 
men receive  per  week  ? 

38.  A  street  commissioner  had  347  men  at  work.  At  uniform 
wages  the  weekly  pay  roll  amounted  to  $2602.50.  What  did  each 
man  receive  per  day  ? 

Six  workmen  received  pay  for  26  days'  work  as  follows  : 
39.  The  carpenter,  $55.25.      40.  Painter,    $52.         41.  Bricklayer,  $65. 
42.  Plumber,  $68.25.      43.  Laborer,  $42.25.     44.  Plasterer,     $63.70. 

What  were  the  daily  wages  of  each  ? 


118  STANDARD  ARITHMETIC. 

Miscellaneous    Examples. 

1.  Mr.  Jacobs  paid  me  $27.34  ;  Mr.  Niel,  $79.14 ;  Mr.  French, 
$34.27;  Mr.  Myers,  $647.79.  My  expenses  on  the  tour  of  col- 
lection were  $19.68.  When  I  started  out  I  had  $50.75  in  my 
purse.     How  much  money  ought  I  to  have  had  on  my  return. 

2.  In  1873  I  paid  $52.23  taxes  ;  in  1874,  $50.79  ;  in  1875, 
$46.27  ;  in  1876,  $44.83  ;  in  1877,  $42.21  ;  and  in  1878,  $40.90. 
How  much  in  these  6  years  ?  The  repairs  on  my  house  in  the 
meantime  cost  $238. 65.  For  the  first  3  years  I  received  $650  per 
year  rent,  for  the  last  3  years  $700  per  year.  How  much  did  I 
receive  in  the  6  years  clear  of  expenses  ? 

3.  A  lady  had  $30.  She  bought  a  dress  for  $9.15,  shoes  for 
$3.40,  a  bonnet  for  $4.50,  and  23  yd.  muslin  at  25<f  a  yd.  How 
much  did  she  have  left  ? 

4.  A  farmer  owed  $500,  and  gave  in  part  payment  435  bu. 
wheat,  at  $1.02  a  bu.     How  much  money  was  yet  due  ? 

5.  If  you  spend  $0.74  a  day,  how  much  will  you  save  in  a  year 
if  your  salary  is  $475  ?    (365  days  to  the  year.) 

6.  If  a  laborer  works  60  days,  10  hours  a  day,  and  receives 
$135  for  his  labor,  how  much  does  he  earn  per  hour  ? 

7.  Mr.  Jordan  sells  Mr.  Marsh  the  following  articles  :  3  cash- 
mere long  shawls,  at  $45.75  each  ;  45  yd.  black  satin,  at  $2.20  a 
yd.;  12  alpaca  umbrellas,  at  $1.35  apiece;  60  worsted  cord  and 
tassels,  at  $0.35  apiece.     Find  the  cost. 

Note. — In  the  following  memoranda  the  price  per  pair,  yard,  doz.,  or  single 
article  is  given  as  usually  written.  The  pupil  is  required  to  extend  the  items,  and 
find  the  footings.    The  sign  @  stands  for  the  word  at  (that  is,  per  pair,  per  lb.,  etc.). 

8.  Find  the  cost  of  9.  How  much  must  be  paid  for 
3  pr.  kid  gloves,            @,  $1.35  5  gal.  vinegar,       @.  $.27 


7  yd.  Malta  lace, 

@ 

.76 

15  lb.  cheese, 

@ 

.09 

4  doz.  handkerchiefs, 

® 

1.80 

3  lb.  hominy, 

@ 

.45 

1  doz.  pair  linen  cuffs. 

,@ 

.13 

27  lb.  sugar, 

@ 

.07 

6  yd.  silk  fringe, 

@ 

1.10 

22  lb.  soap, 

@ 

.06 

3  ostrich  plumes, 

@ 

3.25 

3  gal.  molasses, 

% 

.70 

3  pr.  linen  gloves, 

@ 

.65 

6  lb.  prunes, 

% 

.16 

UNITED  STATES  MONEY.  119 

10.  If  you  are  sent  with  a  $2  bill  to  the  bakery  to  get  1  doz. 
rolls,  at  10  apiece ;  3  loaves  of  bread,  at  80  a  loaf ;  3  doz.  cookies, 
at  10^  a  doz. ;  and  100  worth  of  caramels,  what  change  will  you 
bring  back  ? 

n.  Mother  makes  the  following  purchase  at  the  crockery  store: 
1  cream  pitcher,  750 ;  1  sugar  bowl,  450  ;  1/2  doz.  plates,  at  $1.20 
a  doz. ;  y>  doz.  egg  cups,  at  900  a  doz. ;  3  cups  and  saucers,  at  600 
a  pair  ;  2  doz.  fruit  jars,  at  90  apiece.     What  is  the  bill  ? 

12.  If  your  mother  sends  you  to  the  grocery  with  $5  to  buy 
%  lb.  of  tea,  at  900  a  lb.  ;  1  lb.  of  coffee,  at  400  a  lb. ;  5  lb.  of 
granulated  sugar,  at  110  a  lb. ;  3  lb.  of  lump  sugar,  at  120  a  lb. ; 
1  small  bag  of  salt,  at  90  ;  4  loaves  of  bread,  at  60  a  loaf ;  1  peck 
of  apples,  at  800  a  bu.,  what  change  will  you  receive  ? 

13.  If  you  are  sent  with  $2  to  buy  3  lb.  rice,  at  100  a  lb.  ; 
20  lb.  of  flour,  at  50  a  lb. ;  6  lb.  of  cheese,  at  90  a  lb. ;  5  lb.  of 
prunes,  at  160  a  lb. ;  1  gal.  coal-oil,  at  80  a  quart,  will  you  have 
money  enough  ?  If  not,  which  article  must  you  omit  to  keep  the 
sum  total  within  $2  ? 

14.  Twelve  tons  of  coal  cost  $75.00,  how  much  is  that  per  ton  ? 
What  would  37  tons  cost  at  that  rate  ? 

15.  A  person  sells  5  cows  at  $55  each,  and  a  yoke  of  oxen  at 
$125.  He  agrees  to  take  in  payment  80  sheep.  How  much  do 
the  sheep  cost  him  per  head  ? 

16.  What  is  the  cost  of  39000  feet  of  planed  pine  lumber 
at  $40  per  thousand  feet  ?  Of  16000  shingles  at  $2.75  per  thou- 
sand ? 

17.  Find  the  cost  of  18.  Required  the  cost  of 


60  pr.  overshoes, 

©  $  .65 

9  lb.  lard, 

©  $.08 

15    "   boots, 

©    4.25 

15  lb.  butter, 

©    .22 

17  "    gaiters, 

©    3.85 

18  lb.  pork, 

©    .06 

37   "   slippers, 

©    1.75 

20  lb.  rice, 

©    .09 

230    "   mittens, 

©      .34 

12  lb.  raisins, 

©    .20 

2  doz.  pr.  slippers,  ©    9.00 

4  cans  oysters, 

©    .35 

2  pr.  boots, 

©    9.50 

10  lb.  codfish, 

©    .10 

120  STANDARD  ARITHMETIC. 

19.  Mr.  White  bought  6  tubs  of  butter,  containing  58  lb. 
each,  for  $80.04.     How  much  did  he  pay  per  lb.  ? 

20.  He  sold  the  butter  at  a  profit  of  \%f  a  pound.  Deduct- 
ing 7  lb.,  which  he  used  in  his  family,  how  much  did  he  get 
for  it  ? 

21.  When  coal  is  $4%  per  ton,  how  many  tons  can  be  bought 
for  $238  ?    How  much  would  be  saved  by  buying  at  $4  per  ton  ? 

22.  A  farmer  buys  goods  amounting  to  $235.75.  He  pays  in 
cash  $58.25,  and  agrees  to  pay  the  balance  in  rye,  at  $1.25  a 
bushel.     How  many  bushels  will  be  required  ? 

23.  How  many  pounds  of  cheese,  at  15^  a  lb.,  must  be  given  in 
exchange  for  14  yd.  of  gingham,  at  30^  a  yard  ? 

24.  Subtract  $37.87  from  $237.37  ;  from  the  remainder  sub- 
tract $37.87,  and  continue  subtracting  till  the  remainder  is  less 
than  the  subtrahend.  What  is  the  remainder  ?  Is  this  the  short- 
est way  to  find  the  remainder  ? 

25.  Multiply  $4.35  by  2  ;  multiply  the  product  by  3  ;  multiply 
the  second  product  by  4  ;  the  next  by  5  ;  the  next  by  6  ;  and  the 
next  by  7.     What  is  the  last  product  ? 

A  r range  each  line  in  column,  and  add : 

26.  $13.44,   $300,   $55.25,   $288.39,   $19.50,   $31.67,   $509.07. 

27.  $67.31,  $180.61,  $79.03,  $152.70,  $14,23,  $11.12,  $50.22. 

28.  $88.75,  $264.16,  $44.56,  $76.82,  $30.50,  $72.39,  $142.33. 

29.  $10.13,   $7.56,   $2.18,   $55.44,   $11.19,    $70.25,   $312,  $9. 

30.  $13.33,    $72.69,    $15,437,    $34,805,     $125,595,    $77,666. 

31.  Add  together  all  the  sums  of  money  given  in  examples  26 
to  30,  inclusive. 

•  32.  A  store-keeper,  who  was  about  to  pay  some  debts,  found 
that  he  had  $37.45  in  change  and  $76  in  bank-notes  in  his  money- 
drawer,  $318  in  his  safe,  and  $98.36  in  his  pocket-book.  How 
much  had  he  left  after  paying  5  bills  of  $56.10,  $38.05,  $48.00, 
$213,  and  $78.90,  respectively  ? 


UNITED  STA  TES  MONEY.  121 

33.  The  treasurer  of  a  street  railroad  took  400  dimes,  800 
quarter-dollars,  23  twenty-cent  pieces,  600  half-dollars,  1000  five- 
cent  pieces  to  be  exchanged  for  $5  bills.     How  many  did  he  get  ? 

34.  A  grocer  exchanged  bills  for  small  change.  How  many  50 
pieces  could  he  get  for  $5,  $10  ?— How  many  dimes  for  $5,  $10  ? 
— How  many  quarters  for  $25,  $45  ? — How  many  half-dollars  for 
$37,  $54,  $96  ? 

Making  Change. 

96.  l.  You  buy  three  pounds  of  rice  at  90  a  pound,  and  hand 
the  grocer  in  payment  a  dollar  bill ;  how  does  he  count  the 
change  due  you  ? 

Answer. — Giving  you  the  rice  he  would  count  that  as  27^ ;  then  placing  in  your 
hand  successively  3^,  10^,  10^,  and  50^,  he  would  count  30,  4.0,  50  $1.  This 
is  the  most  convenient  way,  and  least  liable  to  error.  It  is  similar  to  the  "  making 
up"  method  in  subtraction,  which  is  recommended  on  page  41. 

2.  Having  only  50  and  100  pieces,  how  will  the  change  be 
counted,  taking  150  out  of  $1  ? 

3.  Having  10,  50,  100  pieces,  and  $1  bills,  how  would  you 
make  the  change  for  350  out  of  $2  ?  For  850  out  of  $5  ?  For 
75^  out  of  $10  ?  For  $1.12  out  of  $1.50  ?  For  $6.03  out  of  four 
two-dollar  bills  ?    For  670  out  of  $2  ?    For  $3.33  out  of  $5  ? 

4.  If  the  merchant  has  no  change  except  250,  500,  and  $1 
pieces,  how  can  he  make  change  for  $2.75  out  of  $5  ? 

5.  A  collector  presents  a  bill  for  $1.90  ;  you  have  only  two  $1 
bills,  one  100,  and  one  50  piece  =  $2.15.  The  collector  has  only 
large  bills  and  quarter  dollars.  How  can  the  change  be  made  ? 
(If  you  were  to  give  him  your  $2.15,  could  he  then  make  the  change?) 

6.  You  owe  $2.75,  but  have  only  three  dollars  and  a  quarter. 
How  can  change  be  made,  the  collector  having  none  less  than  a 
half  dollar  ? 

7.  If  a  grocer  has  only  small  change,  namely,  10,  20,  30,  50, 
100,  200,  250,  500  pieces,  how  can  he  make  change  for  $2,  the 
goods  you  have  bought  costing  160  ? 


122  STANDARD  ARITHMETIC. 

8.  If  a  merchant  has  only  $1,  $2,  and  $5  bills,  and  l<f  and  5^ 
pieces,  how  will  the  change  for  $3.27  be  counted  out  of  a  $20 
note  ? 

9.  Having  10,  5^,  25 <f,  and  50^  pieces,  and  $2  bills,  how  can 
you  count  the  change  for  $1.15  out  of  $5  ?  For  $2.23  ?  For 
$3.45  ?  For  $1.84  ?  For  $4.66  ?  For  $1.11  ?  For  93^  ?  For 
$4.46  ? 

10.  Having  only  l<fi,  10^,  25^,  and  $1  pieces,  how  can  yon 
count  the  change  for  27^  out  of  $2  ?    For  $1.34  out  of  $5  ? 

Count  out  the  proper  change  in  each  of  the  following  trans- 
actions : 

Goods  sold.  Money  received. 

11.  5  lb.  coffee,  @  320 ;  5  lb.  sugar,  @,  90 ;  3  lb.  cheese,  @  140.    $5 

12.  2  lb.  beef,  @  190;  2  lb.  butter,  @,  420;  radishes,  100.  2 

13.  3  lb.  soap,  @  100;  2  lb.  starch,  @.  120;  1  paper  allspice,  120.    1 

14.  V2  lb.  tea,  @  90 ;  4  lb.  sugar,  @  110 ;  1  qt.  strawberries,  250.      2 

15.  1  lead-pencil,  100;  envelopes,  100;  1  daily  newspaper,  30.        5 

16.  5  yd.  muslin,  @,  130 ;  3  spools  cotton,  @,  60 ;  6  handker- 

chiefs, @  280.  4 

17.  900  lb.  pork,  @.  30 ;   5  bu.  peaches,  @  $2.50 ;  Y  bu.  apples, 

@  750.  50 

18.  18  bu.  oats,  @,  500;  12  bu.  corn,  @,  750;  13  cwt.  hay,  @, 

$1.05.  35 

19.  3  pk.  peaches,  @,  $2  per  bu. ;  5  qt.  cherries,  @  180;  1V2 

lb.  butter,  @  480.  5 


CHAPTER    VIII. 

FACTORS    AND    DIVISORS. 

Definition. 

97.  An  Integer  is  a  whole  number.  It  is  so  called  to  distin- 
guish it  from  a  fraction.      (This  chapter  treats  of  integers  only.) 

i.  2.  s.  4.  i  e.  7.  s.  9.  Factors. 

3 ;  ;;;;;.*;  \  Having  counted  by  8's  and  9's  to  72,  the 

4 pupil  learned  that 

e. 8  times  9=72,  and 

'• 9     u      8=72. 

8.  •••••••*•  • 

Then  he  learned — 

1.  That  8  and  9  are  called  factors  of  72. 

2.  That  72  is  called  the  product  of  8  and  9  ;  and 

3.  That  72  is  called  a  multiple  of  9  ;  also  of  8. 

98.  But  since  8  times  9=72  and  9  times  8=72,  nine  is  con- 
tained exactly  8  times,  and  8  exactly  nine  times  in  72.  Thus 
the  factors  of  a  number  are  exact  divisors  of  that  number. 

99.  Hence,  to  find  the  factors  of  a  given  number  we  ascertain 
by  trial  what  numbers  will  divide  it  without  a  remainder;  the 
divisors  and  quotients  are  the  factors  sought. 

Thus  we  obtain  7  different  pairs  of  factors  of  210,  as  follows  : 
2)210  3)210  5)210  6)210  7)210  10)210  14)210 

105  70  — 42  ,85  ~80  21  15 


121         •  STANDARD  ARITHMETIC. 

Definitions. 

100.  A  Factor  of  a  number  is  any  one  of  two  or  more  in- 
tegers which,  multiplied  together,  produce  the  number. 

101.  When  one  number  can  be  divided  by  another  without 
remainder,  the  dividend  is  said  to  be  divisible  by  the  divisor, 
and  the  divisor  is  called  a  Measure  or  Exact  Divisor  of  the 
dividend. 

102.  A  number  that  is  the  product  of  other  factors  besides 
itself  and  one  is  called  a.  Composite  Number. 

Note  1. — A  Composite  number  is  so  called  because  it  is  composed  of  other 
factors. 

Note  2. — Since  a  composite  number  is  divisible  by  its  factors,  it  may  be  defined 
to  be  a  number  that  is  divisible  by  other  numbers  besides  itself  and  one. 

103.  A  Prime  Number  is  one  that  has  no  factors  and  hence 
no  exact  divisor  except  itself  and  1. 

1 04-.  A  Prime  Factor  is  a  factor  which  is  a  prime  number. 

105 .  An  Even  Number  is  one  that  is  divisible  by  2. 

106.  An  Odd  Number  is  one  that  is  not  divisible  by  %. 


SLATE     EXERCISES. 

1.  Write  in  columns  the  numbers  in  order  from  1  to  35  ;  also 
from  36  to  70,  inclusive,  and  opposite  to  each  write  all  the  pairs 
of  factors  that  will  produce  it.     Thus 

1=1x1  36=2x18,  3x12,  4x9,  6x6 

2=1x2  37=1x37 

3  =  1x3  38=2x19 

4=2x2  39=3x13 

5  =  1x5  40=2  x  20,  4x10,  5x8 

6=2x3  41  =  1x41 
etc.                     .  etc. 

2.  In  the  same  manner  write  the  pairs  of  factors  of  numbers 
from  71  to  107,  inclusive  ;  also  from  108  to  144. 

3.  Make  a  table  such  as  the  one  required  in  Ex.  5,  p.  53, 
omitting  the  first  line  and  first  column.     Give  the  factors  orally. 


FACTORS  AND  DIVISORS.  125 

<L  Make  a  separate  list  of  numbers  from  1  to  144  that  have 

2  for  one  or  more  of  their  factors.  Notice  that  the  right-hand 
figure  of  each  is or  — .  (?) 

5  for  one  or  more  of  their  factors.  Notice  that  the  right-hand 
figure  of  each  is  —  or  — .  ( ?) 

3  for  one  or  more  of  their  factors.  Divide  the  sum  of  the 
digits  of  each  of  these  numbers  by  3,  and  notice  the  remainder, 
if  any. 

9  for  one  or  more  of  their  factors.  m  Divide  the  sum  of  the 
digits  of  each  by  9,  and  notice  the  remainder,  if  any. 
Thus  we  discover  some 

AIDS     IN     FINDING     FACTORS. 

107.  It  may  be  shown  to  be  true  of  any  number  that  it  has 

2  for  a  factor  if  the  right-hand  figure  is  2,  4,  6,  8,  or  0  ; 
5  for  a  factor  if  the  right-hand  figure  is  0  or  5  ; 

3  for  a  factor  if  the  sum  of  its  digits  is  divisible  by  3  ; 
9  for  a  factor  if  the  sum  of  its  digits  is  divisible  by  9. 


Apply  the  foregoing  aids  in  the  following  exercises  : 

l.  Tell  which  of  the  dividends  at  the  top  of  p.  82  are  divisible 

by  2  ;  by  5  ;  by  3  ;  by  9. 

.  2.  Write  10  numbers  of  three  or  more  figures  each,  all  of 

which  shall  be  divisible  by  2  ;  by  5  ;  by  3  ;  by  9. 

3.  Change  one  figure  in  each  of  the  following  numbers,  so  as 

to  make  the  number  divisible  by  2  :   (State  what  change  you  make,  and  why.) 
4379  6479  5243  7957  4343 

5627  8123  2147  8971  5557 

8291  4871  9281  3629  4441 

4.-6.  Change  one  figure  in  each,  so  that  the  number  shall  be 
divisible  by  5. — Change  the  last  figure  in  each,  so  that  the  num- 
ber shall  be  divisible  by  3. — Change  the  first  figure  in  each,  so 
that  the  number  shall  be  divisible  by  9. 


126  STANDARD  ARITHMETIC. 

Factoring. 

108.  Since  7  is  a  factor  of  14,  it  must  be  a  factor  of  any  num- 
ber of  times  14,  as  28,  42,  etc.     Thus  it  is  true  always  that 

A  factor  of  a  factor-of-a-number  is  a  factor  of  the  number 
itself. 

Hence,  having  obtained  one  prime  factor  of  a  number  directly 
from  the  number  itself,  a  second  one  may  be  obtained  from  the 
quotient  of  the  first,  and  a  third,  if  any,  from  the  quotient  of 
the  second,  etc.     Thus 

Solution.  Explanation. — 2  being  contained  105  times  in 

210,  2  and  105  are  factors  of  210.  Then  dividing 
the  quotient  by  3,  we  find  that  3  and  35  are  factors 
of  105,  and  hence  also  of  210;  and,  again,  finding 
that  5  and  7  are  factors  of  35,  we  know  that  they 
are  factors  of  105  and  also  of  210.     Thus  we  de- 

210  r*ve  *^e 

109.  Hule. — Divide  the  given  number  by  any  prime  factor,  and 
if  the  quotient  is  not  a  prime  number,  divide  it  in  like  manner,  and 
so  continue  to  divide  till  the  quotient  is  a  prime  number.  The 
divisors  and  the  last  quotient  are  the  factors  sought. 


2 
3 

210 
105~ 

5 

35 

1 
2x3x 

7 
'roof, 
5x7= 

SLATE     EXERCISES 

Find  the  prime  factors  of 


1.  1050 

6.  5985 

11.  8140 

16.  1906 

21.  2526 

2.  2625 

7.  4620 

12.  8712 

17.  1858 

22.  2978 

3.  1820 

8.  4802 

13.  1320 

18.  1478 

23,  2992 

4.  1485 

9.  5432 

14.  1768 

19.  2956 

24.  3936 

5.  1155 

10.  7000 

15.  1848 

20.  2406 

25.  3430 

Note. — The  learner  will  avoid  useless  labor  if  he  will  keep  it  in  mind  that  the 
quotient  is  as  much  a  factor  of  the  dividend  as  the  divisor  itself. 

Suppose,  for  instance,  that  he  is  working  to  find  the  prime  factors  of  479,  as 
in  the  last  of  the  preceding  examples.  He  tries  successively  every  prime  number 
from  2  upward,  till  he  comes  to  23,  when  he  finds  that  the  quotient  has  become  less 
than  the  divisor.  Here,  if  he  stops  to  think,  he  will  say  to  himself :  "  It  is  of  no 
use  to  try  any  further.  This  number  can  not  have  an  exact  divisor  greater  than 
23,  for  if  it  had,  it  would  have  another  less  than  23  ;  but  I  have  tried  every  prime 
number  from  2  to  23,  and  I  know  it  has  none.     This  number  is  prime." 


FACTORS  AND  DIVISORS.  127 

Common  Factors  and  Common  Divisors. 

110.  A  factor  that  occurs  in  each  one  of  two  or  more  num- 
bers is  a  Common  Factor  of .  those  numbers.  Thus  15=8x5  and 
21=3  x  7 ;  the  factor  3  is  common  to  15  and  21. 

Note. — A  factor  is  said  to  be  common  to  two  or  more  numbers,  just  as  we  may 
say  that  the  letter  i  is  common  to  all  the  syllables  of  the  word  Mississippi. 

111.  A  factor  of  a  number  being  an  exact  divisor  of  the  num- 
ber, a  factor  that  is  common  to  two  or  more  numbers  is  a  common 
divisor  of  those  numbers. 

Find  the  prime  factors  of  252  and  2310,  and  show  that  the  common  factors  are 
common  divisors  of  those  numbers. 

112.  The  product  of  any  two  or  more  prime  factors  of  a  num- 
ber being  an  exact  divisor  of  that  number,  the  products  of  the 
prime  factors  common  to  two  or  more  numbers  are  common  divi- 
sors of  those  numbers. 

Show  that  the  products  of  the  prime  factors  that  are  common  to  294  and  315 
are  common  divisors  of  those  numbers. 

113.  Since  an  exact  divisor  of  a  number  can  have  no  factor 
which  does  not  occur  in  that  number,  the  product  of  all  the 
prime  factors  that  are  common  to  two  or  more  numbers  is  the 
gre&test  common  divisor  of  those  numbers. 

Find,  if  you  can,  any  other  divisors  of  396  besides  its  prime  factors,  and  the 
products  of  two  or  more  of  them.  Can  1820  and  2310  have  any  other  common 
divisors  than  their  common  factors  and  their  products  ?     Try  to  find  one. 

M4-.  Hence,  to  find  the  greatest  common  divisor  of  two  or 
more  numbers — 

Mule. — 1.  E-esolve  the  given  numbers  into  their  prime  factors. 
2.  Multiply  together  all  the  factors  that  are  common  to  all  the 
numbers.    The  product  will  be  the  greatest  common  divisor  sought. 

Example.— Find  the  g.  c.  d.  of  546  and  910  ? 

Prime  Factors  Found.  Prime  Factors  Arranged. 


546 
878 

2 
5 
7 

910 
455 

540=2x3x7x13 
910=2x5x7x13 

91 

91 

Common  Prime  Factors  Multiplied. 

13 

13 

2x7x13=152  g.  c.  d. 

2 

Opera 
924 

tion. 
990 

3 
11 

462 
154 

495 
165 

14 

15 

128  STANDARD  ARITHMETIC. 

115.  A  shorter  Process. — When  the  prime  factors  are  readily 
detected,  the  process  is  somewhat  sim- 
plified and  considerably  shortened  by  di- 
viding the  given  numbers  only  by  the 
factors  that  are  common,  and  multiply- 
ing these  together  for  the  greatest  com- 
2x3x11=00  g.  c.  d.  mon  divisor.  The  principle  of  this  op- 
eration is  the  same  as  that  of  the  one 
given  at  the  bottom  of  the  preceding  page. 

Note. — The  product  of  any  two  of  the  common  prime  factors  is  a  common 
divisor,  but  it  requires  the  continued  product  of  all  of  them  to  make  the  greatest 
common  divisor. 

ORAL     EXERCISES. 

Find  the  greatest  common  divisor  of 

i.  14  and  21               6.  57  and  69              11.  32  and  64  16.  46  and  28 

2.  26  and  39              7.  15  and  93              12.  34  and  38  17.  36  and  54 

3.  40  and  56              8.  50  and  75              13.  58  and  87  18.  49  and  98 

4.  72  and  99              9.  56  and  84              14.  18  and  82  19.  54  and  81 

5.  80  and  48             10.  21  and  56              15.  81  and  45  20.  63  and  81 


SLAT  E     EXERCISES. 

Find  the  greatest  common  divisor  of 

21.  323  and  425                    26.  7008  and  7968  31.  7992  and  9900 

22.  228  and  399                    27.  7568  and  3784  32.  8100  and  6300 

23.  615  and  735                    28.  3876  and  1983  33.  9864  and  9528 

24.  819  and  945                    29.  7956  and  7668  34.  6144  and  6930 

25.  949  and  871                    30.  7378  and  9758  35.  4374  and  5508 

•  Find  the  greatest  common  divisor  of 

36.  45,  57,  and    81               38.  36,  54,  and  56  40.  306,  408,  and  510 

37.  63,  99,  and  126               39.  72,  84,  and  90  41.  420,  462,  and    84 


Find  the  largest  number  that  will  exactly  divide 

42.  546,  462,  and    882  44.  19635,    5355,  8925,  and  12495 

43.  900,  936,  and  2520  45.  19782,  16485,  and  14287 


FACTORS  AND  DIVISORS. 


129 


Cancellation. 

116.  The  principle  that  dividing  divisor  and  dividend  by  the 
same  number  does  not  alter  the  quotient  may  be  demonstrated  as 
follows  : 

Explanation.  —  Any  divisor  and  dividend  having  a 
common  factor  may  be  represented  by  an  equal  number 
of  rows  of  dots,  as  18  and  54  at  the  right.  Whence  it 
becomes  evident  that  any  part  of  the  divisor  as  one  or 
more  lines  is  contained  in  a  like  part  of  the  dividend,  as 
many  times  as  the  whole  divisor  is  contained  in  the  whole 
dividend. 


Divisor.       Dividend. 


117.  Hence  any  factor  or  number  of  factors 
common  to  divisor  and  dividend  may  be  rejected  without  affecting 
the  quotient. 

That  this  principle  may  be  well  impressed  upon  the  mind,  let  many  examples, 


Example. — l.  (a)  Divide  the  product  of  7,  3,  13,  5,  7,  19,  and 
3,  by  the  product  of  7,  3,  7,  5,  and  3.  (b)  Reject  all  common 
factors,  find  the  product  of  the  remaining  ones  and  divide. 

Note. — The  factor  1  always  remains  in  place  of  factors  omitted.  It  is  not 
written  because  it  does  not  affect  the  result. 

Example. — 2.  How  many  bushels  of  oats  at  48^  a  bu.  can  Mr. 
A.  get  for  3  crocks  of  butter  containing  8  lb.  each  at  32^  a  lb. 

Analysis.— At  320  a  pound,  8  lb.  of  butter  will 
cost  8  times  320  =  2650,  and  3  crocks  containing  8 
lb.  each  will  cost  3  times  2560  =  $7.68,  and  as 
many  bushels  can  be  bought 
for  $7.68  as  there  are  times  $ 

480  in  $7.68  =  16. 

But  we  may  indicate  this 
work  by  writing  the  factors 
(makers)   of   the  dividend  on 

the  right  and  the  divisor  on  the  left   side  of  a  vertical 

line,  and  shorten  the  work  by  rejecting  common  factors,  as  shown  at  the  right. 

118.  To  cancel  is  to  erase  or  cross  out,  hence  the  word  Can- 
cellation is  applied  to  erasing  or  crossing  out  factors  and  terms 
which  counterbalance  each  other  in  an  arithmetical  operation. 


Solution. 

32<f 

16 

8 

48^)768^ 

256$* 

48 

3 

288 

$7.68 

288 

n  2 
$ 
j 

16  Ans. 


130  STANDARD  ARITHMETIC. 


l. 


SLATE     EXERCISES. 

15X57X36X35  ~    42x18x65x11 


12X19X25  35X45X13 

17X95X8X23  91x36x94x54 


85X38X5  78X14X18 

5.  How  many  cheeses,  weighing  49  lb.  each,  at  120  a  pound, 
must  be  given  in  exchange  for  13  barrels  of  flour  at  40  a  pound  ? 

(196  lb.  =  1  barrel  of  flour.) 

6.  How  many  sacks  of  wheat,  containing  3  bu.  each,  at  960  a 
bushel,  must  be  given  for  78  sacks  of  potatoes,  each  containing 
2  bu.  at  640  a  bu.  ? 

7.  A  lady  bought  9  yards  of  ribbon  at  560  per  yard,  but  ex- 
changed it  for  other  ribbon  at  320  per  yard ;  how  many  yards 
did  she  then  get  ? 

8.  At  1129  for  27  acres  of  land,  what  will  180  acres  cost  ? 

9.  At  what  price  per  yard  will  5  bales  of  cloth,  containing  12 
pieces  of  42  yards  each,  pay  for  50  rolls  of  carpeting,  of  75  yards 
each,  at  $2. 10  per  yard  ? 

10.  Divide  34x102x85  by  51x17X68. 

11.  Divide  16773x13401x11412  by  11182x11912x8559. 

How  many 

12.  Bu.  apples  @  600  will  pay  for  35  lb.  tea  @  840? 

13.  Bu.  peaches  %  $3.50  "  "  "  25  tons  coal  @  $12  60? 

14.  Pieces  muslin  (39  yd.)  %  120  "  "  "   26  tubs  butter  (72  lb.)  @  320  ? 

15.  Tons  hay  ©  $12.18  "  "  "   3  barrels  sugar  (232)  @  90? 

16.  Horses  @  $91  "  u  "   7  acres  land  @  $559  ? 

17.  Cows  @  $39  "  "  "   650  sheep  @  $3? 

18.  Mules  @  $125  "  "  "   25  horses  @t  $160? 

19.  Tubs  butter  (54  lb.)  %  280  "  "  "   378  yd.  muslin  @  160? 

At  what  price  will 

20.  65  lb.  coffee  pay  for  26  bu.  potatoes  @  450? 

21.  75  acres  land  "      "   21  horses  @.  $125  each? 

22.  26  barrels  pork  "      "   78  bl.  flour  @  $6? 

23.  260  doz.  eggs  "      "   78  yd.  silk  ©  900? 


FACTORS  AND  DIVISORS.  131 

Factors  and  Multiples. 

119.  An  integral  (or  whole)  number  of  times  a  number  is  a 
multiple  of  that  number.     (See  note,  p.  53.) 

Note. — A  multiple  of  a  number  being  some  whole  times  that  number,  is,  of 
course,  always  divisible  by  it ;  hence  a  multiple  of  a  number  is  sometimes  defined 
to  be  "  a  number  that  is  exactly  divisible  by  it." 


SLATE     EXERCISES. 

l.  Write  in  columns  the  multiples  from  1  to  120  of  3,  of  4, 
of  5,  of  6,  of  7,  and  of  8.     Thus, 


3 

4 

5 

6 

7 

8 

6 

8 

10 

12 

14 

16 

9 

12 

15 

18 

21 

24 

12 

16 

20 

24 

28 

32 

15 

20 

25 

30 

35 

40 

18 

24 

30 

36 

42 

48 

etc. 

etc. 

etc. 

etc. 

etc. 

etc. 

2.  Make  a  list  of  the  numbers  from  1  to  120  that  are  mul- 
tiples of  both  3  and  4  ;  thus — 

12,  24,  36,  48,  60,  72,  84,  96,  108,  120. 

3.  In  like  manner  make  a  list  of  the  multiples  from  1  to  120 
that  are  common  to  3  and  5,  3  and  6,  etc. 

4.  Make  a  list  of  the  multiples  from  1  to  120  that  are  common 
to  3,  4  and  5  ;  3,  5  and  6,  etc.,  as  far  as  may  be  directed. 

5.  Make  a  list  of  the  multiples  that  are  common  to  3,  4,  5  and 

6;  to  4,  5,  6  and  7  ;  to  5,  6,  7  and  8,  as  above. 

Note. — The  pupil  should  note  the  fact  that  any  number  may  have  an  unlimited 
number  of  multiples,  and  that  any  two  or  more  numbers  may  have  an  unlimited 
number  of  common  multiples,  but  only  one  least  common  multiple. 

120.  Any  multiple  of  a  number  must  contain  at  least  all  the 
prime  factors  of  that  number. 

Thus  2  and  3  being  factors  of  6,  they  must  occur  as  factors  in  any  number  of 
times  6.     [3  times  6  is  3  times  (2  times  3),  and  so  on,  to  any  number  of  times  6.] 
Though  the  line  of  marks  at  the  right  should  be  copied  millions  of      ////// 
times,  the  number  of  marks  could  never  escape  the  factors  2  and  3.     ////// 


132  STANDARD  ARITHMETIC. 

121.  A  multiple  that  is  common  to  two  or  more  numbers  must 
contain  at  least  all  the  prime  factors  that  enter  into  each  of  them. 

Thus,  any  multiple  common  to  15  and  21  must  contain  the  factors  3,  5  and  7, 
for  no  number  that  does  not  contain  the  factors  3  and  5  can  be  a  multiple  of  15, 
and  no  number  that  does  not  contain  the  factors  3  and  7  can  be  a  multiple  of  21. 

Note. — A  common  multiple  of  15  and  21  may  contain  any  other  factors  besides 
3,  5,  and  7,  but  these  it  must  contain. 

122.  The  least  common  multiple  of  two  or  more  numbers  must 
contain  all  the  prime  factors  that  enter  into  each  of  them,  and 
no  others. 

Thus,  18  and  24  being  resolved  (separated)  into  their  prime  factors,  we  have 

18=2x3x3 
24=2x2x2x3. 

Hence,  any  common  multiple  of  18  and  24  must  contain  the  factors  2,  2,  2,  3  and 
3,  and  the  least  common  multiple  must  contain  no  other.  If  it  did  contain  any  other 
factor  it  would  not  be  the  least  common  multiple. 


ORAL    EXERCISES 

1.  What  is  the  least  common  multiple  of  9,  14,  and  21  ? 

Oral  Solution. — 1.  A  multiple  of  21  must  contain  the  factors  3  and  7.    3  x  7=21. 

2.  A  multiple  of  14  must  have  the  factors  2  and  7,  hence  the  common  multiple 
of  21  and  14  must  contain  3,  7,  and  2  as  factors.     3  x  7  x  2=42. 

3.  A  multiple  of  9  must  have  the  factors  3  and  3,  hence  the  common  multiple 
of  21,  14,  and  9  must  have  the  factors  3,  7,  2,  and  3.     3  x  7  x  2  x  3=226. 

Hence  126  is  the  least  common  multiple  of  9,  14,  and  21,  because  it  contains 
no  factor  which  is  not  necessary  for  one  or  another  of  the  given  numbers. 

2.  Find  the  least  common  multiple  of  6,  7,  9,  12,  14,  18,  21, 

36,  and  42. 

Oral  Solution. — 1.  Since  36  is  a  multiple  of  6,  9,  12,  and  18,  any  multiple  of  36 
will  be  a  multiple  of  these  numbers  also ;  and  since  42  is  a  multiple  of  7,  14,  and 
21,  any  multiple  of  42  will  be  a  multiple  also  of  these  numbers;  hence,  the  least 
common  multiple  of  36  and  42  will  be  the  least  common  multiple  of  all  the  given 
numbers. 

2.  The  factors  of  42  are  2,  3  and  7,  but  the  factors  of  36  are  2,  2,  3  and 
3,  or  one  2  and  one  3,  more  than  are  found  in  42 ;  hence  we  multiply  42  by  2  and 
by  3,  or  at  once  by  6,  and  obtain  the  product  252,  which  is  the  1.  c.  m.  of  all  the 
given  numbers. 


FACTORS  AND  DIVISORS.  133 

Find  the  least  common  multiple 

3.  Of  2,    4,  and    6  7.  Of  2,    3,    4,  and    6         11.  Of  9,  2,    6,  18,  24 

4.  "  3,  4,  and  6  8.  "  4,  8,  12,  and  16  12.  u  8,  7,  12,  21,  24 
6.  "  5,  6,  and  15  9.  "  5,  7,  15,  and  21  13.  "  5,  2,  15,  7,  35 
6.    "   7,  14,  and  21         10.    "   3,  14,  21,  and  28         14.    u   2,  3,    6,    9,  54 

Hence,  for  finding  the  least  common  multiple  of  two  or  more 
numbers,  we  have  the  following 

123.   Utile, — 1.  Resolve  each  number  into  its  prime  factors. 

2.  Take  all  the  prime  factors  of  the  greatest  given  number,  and 
such  factors  of  the  others  as  are  not  found  in  it.  The  continued 
product  of  all  these  factors  will  be  the  least  common  multiple. 

Note. — If  any  of  the  given  numbers  are  multiples  of  others,  the  multiples  only 
need  to  be  considered,  for  the  given  number  that  is  a  multiple  of  another  contains 
all  its  factors. 

SLATE     EXERCISES. 

15.  Find  the  least  common  multiple  of  88,  126,  and  330. 

Explanation. — We  write  out  all 

Solution.  the  factors  of  the  several  numbers, 

gg__2  x  2  x  2  X  11  so  ^at  we  may  readily  see  what 

126  =  3x3x2x7  they  are.     We  then  take  the  factors 

qq0_o  x  q  x  5  x  1 1  of   330,  and   unite  with  them  the 

factors  that  occur  in  the  other  num- 

Least  Common  Multiple.  bers,  and  not  in  330.    The  continued 

2x3x5x11  X3x7x2x  2=27,720        product  of  all  is  the  least  common 

multiple  sought. 
Note. — Any  of  the  given  numbers  may  be  taken  at  once  as  the  product  of  its 
own  prime  factors,  and  this  being  multiplied  successively  by  such  of  the  prime 
factors  of  each  of  the  other  numbers  as  are  not  contained  in  any  preceding  number, 
the  product  will  be  the  1.  c.  m.  sought.     Thus : 

330  x  3  x  7  x  2  x  '2=27,720 

In  like  manner  find  the  1.  c.  m. 

16.  Of  27,  24,  and  15       20.  Of    9,  12,  14,  and  210       24.  19,  27,    36,    63 

17.  "   63,  27,  and  84      21.    "   60,  15,  24,  and    25       25.  13,  17,    19,    32 

18.  "   12,  51,  and  68      22.    "   54,  81,  63,  and    14       26.  23,  27,    54,  108 

19.  "   35,  63,  and  72      23.    "   18,  24,  72,  and  144       27.  14,  17,  105,  110 


28.  Of  9546,  6364,  and  14319  29.  Of  4862,    2002,  and  17017 


134  STANDARD  ARITHMETIC. 

Problems  G.  C.  D.  and  L.  C.  M. 

1.  One  boy's  blocks  are  2  inches  thick,  another's  3,  and  an- 
other's 5.  The  three  boys  build  "  towers  "  of  equal  heights.  How 
high  at  least  are  they  ?    How  many  blocks  does  each  one  use  ? 

2.  What  is  the  least  sum  that  can  be  paid  in  either  2,  3,  5, 
10,  20,  or  25^  pieces  ? 

3.  William  has  27^,  Mary  36^,  and  Harry  51^,  not  in  one-cent 
pieces,  yet  all  in  coins  of  one  denomination.     What  is  it  ? 

4.  In  one  grammar  school  there  are  504  girls,  in  another  there 
are  324  boys.  It  is  desired  to  divide  them  into  classes  of  equal 
size.  How  many  pupils  will  there  be  in  each  class,  if  as  large  as 
it  can  be  made  ? 

5.  The  four  sides  of  a  play-ground  measure,  respectively,  464, 
672,  368,  and  240  ft.  in  length.  How  long  must  the  boards  used 
in  fencing  it  be  cut,  so  that  they  shall  be  of  equal  length,  and  as 
long  as  possible  ? 

6.  On  the  same  day  a  merchant  sends  out  traveling  salesmen 
with  instructions  to  return,  respectively,  in  1,  2,  3,  4,  and  5  weeks. 
When  any  one  returns  he  is  sent  out  again  at  once  for  the  same 
period  as  before.     In  how  many  weeks  will  they  be  together  again  ? 

7.  In  how  many  weeks  would  they  come  in  together  if  sent 
out  for  6,  9,  12,  18,  and  36  weeks,  respectively  ? 

8.  What  is  the  least  sum  a  dealer  in  live  stock  must  have  to  be 
able  to  invest  equal  sums  in  horses  at  $105,  mules  at  $68,  and 
beeves  at  $30  per  head  ?  How  much  if  he  pays  $105  per  head  for 
horses,  $70  for  mules,  and  $30  for  beeves  ? 

9.  A  court-yard  42  ft.  6  in.  long,  and  31  ft.  8  in.  wide,  is  to 
be  paved  with  square  tiles  of  equal  size,  and  as  large  as  possible. 
How  long  and  wide  must  each  tile  be  ? 

10.  A  man  having  on  deposit  $695,  $417,  and  $1251,  respect- 
ively, in  three  different  banks,  wishes  to  draw  out  the  whole  in  as 
large  equal  sums  as  possible.  What  is  the  greatest  sum  for  which 
he  must  draw  his  checks  ? 


CHAPTER   IX. 

FRACTIONS. 
Introductory  Exercises. 

124.  Draw  on  slate  or  paper  twelve  lines  of  equal  length,  and 
about  one  half  inch  apart.  Divide  and  subdivide  the  lines  as 
required  by  the  questions.* 

1.  If  the  first  line  were  divided  into  two  equal  parts,  what 
would  you  call  each  part  ?  How  many  such  parts  in  a  line  ? 
How  many  in  2,  5,  7,  9,  12  lines  ? 

2.  How  many  halves  in  2  lines  and  a  half  ?  In  3  lines  ?  In 
4  lines  and  a  half  ?    In  11  lines  and  a  half  ? 

3.  If  each  half  were  divided  into  two  equal  parts,  how  many  of 
the  new  parts  would  there  be  in  a  whole  line  ?  What  would  you 
call  one  part  ?    Two  parts  ?  etc.    What  part  of  a  half  is  a  fourth  ? 

4.  If  the  other  lines  were  divided  in  the  same  way,  how  many 
fourths  in  each  line  ?    In  3  lines  ?    In  8  lines  ?  etc. 

5.  How  many  fourths  in  2  lines  and  1  fourth  ?  In  5  and  1 
fourth  ?    In  G  and  3  fourths  ?    In  7  and  1  fourth  ?  etc. 

6.  Are  2  halves  less  or  greater  than  a  line  ?  5  fourths  ?  etc. 

7.  How  many  whole  lines  in  2  halves  ?  4  halves  ?  etc.  How 
many  in  3  fourths  ?   5  fourths  ?   9  fourths  ?  etc. 

8.  If  each  fourth  were  divided  into  two  equal  parts,  how  many 
parts  would  there  be  in  a  line  ?    What  would  you  call  them  ? 

(Other  questions  should  here  be  asked,  similar  to  those  on  fourths,  as  above.) 

*  Slips  of  paper  of  any  uniform  length,  paper  squares,  etc.,  etc.,  are  convenient 
materials  for  these  exercises. 


136  STANDARD  ARITHMETIC. 

Draw  12  other  lines.  It  would  be  well  if  these  could  be  just 
12  inches  long. 

1.  If  each  line  were  divided  into  three  equal  parts,  what  would 
the  parts  be  called  ?    Why  ? 

2.  How  would  you  express  1  part  in  figures  ?  2  parts  ?  5  parts  ? 

(For  reading  and  writing  simple  fractions  see  Art.  73,  p.  77.) 

3.  Are  4/3  greater  or  less  than  a  whole  line  ?  How  much  ? 
How  many  thirds  in  3,  5,  7,  12  lines  ? 

4.  How  many  thirds  in  2%  1%  5%,  7V3,  8%  10%? 

5.  How  many  whole  lines,  or  how  many  lines  and  what  parts 
of  a  line,  are  needed  to  make  %  %  %  %,  %  3%,  32/3  ? 

6.  If  each  third  were  divided  into  two  equal  parts,  how  many 
of  the  new  parts  would  there  be  in  a  whole  line  ?  What  would 
you  call  them  ?    How  would  you  express  five  of  them  in  figures  ? 

7.  How  many  of  these  smaller  parts  are  there  in  a  third  ?  In 
2  thirds  ?    In  3  thirds  ?     How  many  sixths  in  3  thirds  ? 

8.  Which  is  greater,  %  or  2/6  of  one  of  these  lines  ?  -Why  ? 
What  part  of  a  third  is  1  sixth  ?    Is  %  a  part  or  the  whole  of  %  ? 

9.  How  many  sixths  in  */,  of  a  line  ?  In  %  ?  In  %  ?  In  2 
lines  ?    In  6,  8,  10,  11  lines  ? 

10.  How  many  sixths  in  1%  ?  In  2%  ?  In  4*/a  ?  In  7%  ? 
In8%?    In8%?    In72/3? 

11.  How  many  lines,  or  lines  and  parts  of  a  line,  are  needed 
to  make  >%,  *%,  %  3%,  s%,  %  %,  •%,  5%  ? 

12.  If  each  sixth  were  divided  into  two  equal  parts,  what  would 
the  new  parts  be  called  ?  How  would  you  express  one  or  more 
of  them  ?  If  the  line  is  12  inches  long,  what  is  the  length  of 
each  part  f 

13.  How  many  of  them  in  %  of  a  line  ?    In  2/3,  %  %,  %  %  ? 

14.  How  many  twelfths  in  2  lines  and  %  ?  In  3  3/12  ?  In  4  5/12  ? 
In7Vi2?    Inl0yi2?     Inlln/12?     How  many  twelfths  in  1% 

2%  4%,  73/12,  9%  lines? 


FRA  CTI0N8. 


137 


Questions  upon  the  Rules  in  the  Margin. 

125.  Note. — The  following  questions  arc  designed  to  be 
only  suggestive  of  exercises  that  may  be  given.  A  foot-rule 
or  a  yard-stick  will  afford  many  others. 

1.  Into  how  many  parts  is  the  first  of  these 
two  measures  divided  by  the  horizontal  line  in 
the  middle  ?  What  do  you  call  the  parts  ?  Into 
how  many  parts  is  the  measure  at  the  right  di- 
vided by  the  longest  horizontal  lines  ?  What  do 
you  call  these  parts  ?    Why  ? 

2.  Which  is  greater,  %  or  y3  ?  How  can  you 
tell  without  seeing  or  measuring  the  parts  ? 

3.  What  parts  of  the  whole  do  you  get  by 
dividing  y3  into  2  equal  parts  ?    Why  ? 

4.  How  many  sixths  in  y3,  %,  3/3  ?  What  part 
of  y2  do  you  get  by  dividing  it  into  2  equal  parts  ? 
What  part  of  the  whole  is  1/2  of  1/2  ? 

5.  How  many  parts  do  you  get  by  dividing 
each  of  the  fourths  into  2  equal  parts  ?  What 
are  these  new  parts  called  ?    Why  ? 

6.  Which  is  longer,  y6  or  y8  ?  Suppose  you 
could  not  see,  nor  measure,  would  you  know 
which  is  the  greater,  3/6  or  3/8  ?    How  ? 

7.  The  sixths  in  the  second  measure  are  di- 
vided each  into  2  equal  parts.  What  is  their 
name  ?    Why  ? 

8.  Are  these  twelfths  as  large  as  the  eighths 
in  the  other  measure  ? 

9.  What  are  the  smallest  parts  of  the  second 
measure  ?    How  many  are  there  ? 

10.  Are  the  smallest  parts  of  the  first  meas- 
ure as  large  as  the  smallest  parts  of  the  second 
measure  ?    Can  you  tell  by  counting  them  ? 


138 


STANDARD  ARITHMETIC. 


ORAL    EXERCISES. 

1.  Explain  how  it  is  that  y2  is  equal  to  %. 

Note. — Divide  any  whole  thing  or  number  into  fourths,  and  show  that  one  half 
is  equal  to  2  fourths. 

2.  Is  %  equal  to  %,  to  4/8,  to  6/12,  to  %  ?    State  why. 

3.  Name  some  other  parts  equal  to  y4 ;  also  parts  equal  to  y6, 

to  y8,  to  y3,  to  y12,  to  %  to  %,  to  %  to  %,  to  %  to  %. 

4.  If  you  had  a  line  divided  into  sixths,  how  could  you  change 
the  sixths  into  twelfths  ?  The  twelfths  back  to  sixths  ? 

5.  How  many  sixths  in  y3,  2/3  ?    How  many  eighths  in  3/4,  */,  ? 

6.  How  many  twelfths  in  %  %  %  %  ?    In  %  %  %  %  ? 

7.  How  many  twenty-fourths  in  3/8  ?  In  5/8?   In  7/8?    How  many 
in  %  %  %  %  ?    How  many  in  %  %  %  ? 

How  much  is 

s.  y2  of  %  ?      9.  V,  of  y.  ?      io.  %  of  %  ? 

%  of  %  ?         >/,  of  %  ?  %  of  'A  ? 

V,  of'/s?         Vsofyls?  y3ofy4? 


11.  VsOf  %? 

%of  y3? 

%of »/,? 


Note. — The  following  representation  of  a  fraction  rule  will  suggest  other  exer- 
cises.    The  figures  at  the  left  show  how  many  parts  each  side  is  divided  into. 


Tl^T^T! 


12.  How  many  wholes  and  ninths  are  in  15/9,  2%,  21/9,  3%,  47/9  ? 

13.  Which  makes  the  larger  parts,  dividing  an  apple  into  lOths 
or  12ths  ?  Which  is  the  greater,  y8  or  %  of  a  thing  ?  %  or  %  ? 
%  or  %  ?    y4  or  %  ? 

14.  Which  is  the  greater,  %  or  %  ?  %  or  %  ?  73/145  or 
29/u5?    5/8or3/8?    yi8or10/18?     Why? 

16.  Draw  two  lines  of  equal  length.  Divide  one  into  thirds, 
the  other  into  fourths,  and  find  how  many  more  twelfths  there 
are  in  y3  than  in  y4. 


FRACTIONS.  139 

Definitions. 

126.  A  Fraction  is  one  or  more  of  the  equal  parts  of  a  unit 
or  whole. 

(27.  The  unit  of  the  fraction  is  the  unit  which  is  divided. 
One  of  the  equal  parts  is  a  fractional  unit. 

(28.  Fractions  obtained  by  the  division  of  the  unit  into  tenths, 
tenths  of  tenths  or  hundredths,  etc.,  are  called  Decimal  Frac- 
tions. All  others  are  called  Common  Fractions,  to  distinguish 
them  from  decimals.         

(29.  Common  Fractions  may  be  expressed  by  words,  as  tivo 
thirds,  or  by  figures,  thus,  %,  the  upper  number  standing  for 
"two,"  the  number  of  parts,  and  the  lower  one  for  "thirds,"  the 
name  of  the  parts.   (See  Art.  73,  page  77.) 

(30.  The  number  of  parts  and  the  name  of  the  parts  are 
called  the  terms  of  the  fraction. 

131.  The  term  which  expresses  the  number  of  parts  is  the 
numerator  (counter  or  numberer).  The  term  which  indicates  the 
name  of  the  parts  is  the  denominator  (namcr). 

Note. — Since  the  denominator  indicates  the  name  of  the  parts  by  showing  how 
many  parts  there  are  in  a  unit,  it  may  be  treated  as  a  number  as  well  as  a  name. 

132.  A  simple  fraction  is  one  whose  terms  are  both  integers, 

^     /9>        /20>    GtC. 

133.  A  proper  fraction  is  one  whose  numerator  is  less  than 
the  denominator,  as  2/3,  3/4,  etc. 

(34.  An  improper  fraction  is  one  whose  numerator  is  equal 
to  or  greater  than  its  denominator,  as  %,  %,  etc. 

135.  A  mixed  number  is  one  which  is  composed  of  an  integer 
and  a  fraction,  as  3y2,  53/7,  etc. 

(36.  An  integer  may  be  expressed  in  the  form  of  an  improper 
fraction  by  writing  it  as  a  numerator,  with  1  as  a  denominator. 

Thus,  5  may  be  written  5/j,  which  is  read  5  ones  or  5. 


Slate  Work. 
67 


140  STANDARD  ARITHMETIC. 

Reductions. 

Changes  of  Form,  not  of  Value. 

137.  To  reduce  integers  or  mixed  numbers  to  improper  fractions, 
and  the  contrary. 

Example.— l.  Eeduce  67%  to  fifths. 

Analysis. — In  1  there  are  6  fifths,  hence  in  67  there  are  5 

67  times  5  fifths=335  fifths;  335  fifths*  8  fifths=338  fifths.  335" 

Note. — The  result  being  the  same,  we  multiply  67  by  5,  ^_ 

as  the  shortest  way  of  obtaining  67  times  5.  338 

Example. — 2.  Reduce  19/5  to  an  integer  or  mixed  number. 

Analysis. — 5  fifths  =  1.    Hence  19  fifths  contain  as  many 
units  as  there  are  times  5  fifths  in  19  fifths  =  34/5.  Slate  Wcrk. 

Suggestion. — For  the  rules  in  these  cases  the  pupil  may  5)19__ 

be  required  to  state  the  processes  by  which  he  obtains  the  34/5 

results. 

Note. — In  oral  exercises  the  pupil  should  be  required  to  announce  results  at 
once  if  possible,  except  when  specially  directed  to  give  an  analysis. 

Reduce  to  improper  fractions 

3.                 4.                  5.  6.                  7.  8. 

1%             2%             5%  4%  1973  31% 

?7a            1%            8%  •    3%  18%  29% 

5%            3%            7%  8%  37%  33% 

Reduce  to  integers  or  mixed  numbers 

9.  10.  11.  12.  13.  14. 

Vi  "A  "/•  ■        "A  3%  "A 


"A  %  15A  "A  "A 

17/  32/  53/  39 


%                         %                       *%                       3%                       6%  3% 

Reduce  mixed  numbers  to  improper  fractions,  and  improper 
fractions  to  integers  or  mixed  numbers. 

15.  28%6           18.  42%           21.  100%           24.  1828/u  27.  487% 

16.  31%3  19.  35%  22.  37510/23  25.  132%3  -        28.  7239/10 

17.  248/17          20.  21%          23.  841  %i          26.  4563/15  29.  8919/13 

30.  How  many  yards  in  35%  of  a  yard  ? 


FRACTIONS.  141 

Reducing  to  Higher  and  Lower  Terms. 

Let  it  be  remembered  that  the  value  of  a  fraction  is  not 
changed  by  multiplying  or  dividing  both  terms  by  the  same  num- 
ber ;  thus — 

4x3_12         •  4-s-2_2 

6x3~ 18  6-*-2~  3 

For  it  is  clear  that     7*\                I                I  I 

is  equivalent  to    ^  |        1        1        I        1 1 1 

or  to  %\  i  i  1  i  i  |  i m  1  i  i  1  ,  ■  |  ■  ■  1 


138.  To  reduce  a  fraction  to  higher  terms  (greater  numerator  and 
denominator). 

Example.—i.  Eeduce  %  to  twelfths. 

Process.  Explanation. — If  each  fourth  of  a  slip  of  paper  be 

g  x  o      q  divided  into  three  equal  parts,  the  whole  slip  will  contain  4 

j  v  q  =  T9  times  3  parts,  or  12  twelfths,  and  3  fourths  will  contain  3 

times  3  parts,  or  9/,  2. 

Hence  the  following 

Utile. — To  reduce  a  fraction  to  higher  terms,  divide  the  required 
denominator  by  the  denominator  of  the  given  fraction,  and  multiply 
both  of  its  terms  by  the  quotient. 


ORAL    EXERCISES. 

Reduce  ■  Reduce  . 

2-  %  %  %  %  to  12ths.  7.  %  %  %  V,2,   Vis,  to  72da. 

3-  1,    %,  %  V*  "    8ths.  8.  %  7e,  %  '%,  14/.8,  ".  54ths. 
*•  Yt>  V*  V„  %  "  18ths.  9.  %  %  '/9,  %    */n,  "  45ths. 

5-  %  %,  %,  'A,    "  24ths.  10.  •/*  %  %,   %   »/„,  "  48ths. 

6-  %  %,  %,  Vu.  "  Wths.  11.  %  %  'As,  "A,   "As,  "  36ths. 

Let  the  first  three  examples  be  illustrated  by  division  of  lines  or  folding  of 
paper. 

12.  Change  %  %  %  %  %  and  %  to  tenths.  ' 

13.  Changed  hundredths  &  %  %*  3/20,  %  %,  %,  %  %  »/* 


U2  STANDARD  ARITHMETIC. 

A 

139.  To  reduce  fractions  to  lower  terms  (smaller  numerator  and  de- 
nominator). 

Note. — In  the  preceding  case  (p.  141)  we  computed  the  numerical  result  of 
dividing  any  given  equal  parts  of  a  thing  or  number  into  smaller  equal  parts.  In 
this  case,  we  are  to  find  the  result  of  uniting  smaller  into  larger  ones. 

Example.— l.  Reduce  %  to  lower  terms. 

Process*  Explanation. — Uniting  each  3  twelfths  of  any  object  into  1 

g|__3/  larger  part,  we  have  4  larger  parts  (fourths),  and  in  the  9  twelfths 

Il2       /4  there  are  3  of  them.     Hence  9/i2  =  3/4- 

Reduce  to  lower  terms 

41/ 
/164 

196/ 
'144  /444 

140.  Thus  we  find  that  fractions  are  reduced  to  lower  terms 
by  dividing  both  terms  by  any  common  factor. 

And,  that  they  are  reduced  to  their  lowest  terms  by  dividing 
them,  successively,  by  all  the  prime  factors  common  to  the  two ; 
or,  by  the  continued  product  of  all,  the  latter  being  their  greatest 
common  factor ,  and  hence  their  greatest  common  divisor. 

141.  When  the  terms  of  a  fraction  are  large,  or  not  readily 
resolved  into  factors,  the  following  method  of  finding  the  greatest 
common  divisor  will  be  convenient. 

4.  Ileduce  475/589  to  lowest  terms. 


2. 

12/ 
/24 

64/ 

In 

/90 

66/ 

/240 

59/ 

/in 

10/ 
/50 

48/ 

In 

15/ 

fn 

% 

3. 

15, 
/35 

6/ 

718 

12/ 
/96 

72/ 
/144 

480/ 
/500 

15/ 
fit 

31/ 

/93 

34/ 

/68 

% 

Process. 


Explanation. — We  divide  the  greater 
'number  by  the  less,  and  the  divisor  by  the 
475)589(1  remainder,  and  so  on  till  there  is  no  re- 

475  mainder.     The  last  divisor  (19)  is  theg,  c.  d. 

114)475(4  sought' 

Why  it  is  that  we  can  always  depend 


456 


19 


on  such  a  process  to  find  the  g.  c.  d.  is  not 


475      25/           19)114(6  readily  understood  by  the  young  learner. 

koq=     /3X               214  ^nc  demonstration  is  therefore  reserved  for 

the  appendix.     No  formal  rule  is  necessary. 

Reduce  to  lowest  terms 

K  1645/        17  1363/          0  8903/         -,-,      1261/        10  1989/ 

B«     /1833      '•     /1739       »•  /13201      »*•     /1649      ld'     /2873 

ft     1589/         p  8903/         ,n  1945/         10  2813/         i  a      ™*5 1 

«=     /2724      o-     /10991      10-  /3501       12.     /3783      14.     /350I 


FRACTIONS.  143 

Addition  of  Common  Fractions. 

ORAL    EXERCISES. 

I.  Mary  takes  %  of  a  pie  for  lunch  at  school,  William  takes  %, 
John  %,  and  Henry  2/6.     How  many  sixths  do  they  all  take  ? 

Find  the  sum  of 
2.    %  +  %  =      3.   %  +  %  =      4.    •/„+  8/u=      5.    4/s  +  %  +>/ _ 

*A  +  %  =  %0+  VlO=  %2+  Vl2=  %  +  V8  +%  = 

%  +  Vs  =  Vi3+  %3=  7n+  Vn=  %  +  %  +%= 

6.  Sarah  has  5/8  of  a  yard  of  ribbon,  and  Lucy  has  % ;  how 
many  8ths  have  they  together  ?  How  many  yards,  and  what  part 
of  a  yard  ? 

Find  the  sum  of 
7.  7%+%*     8.  5%+3/,=     9.  6%  +3  %  =     10.  16  y„+  •/»= 
3%+%=  4%+%=  8V„+BVi.=  U»/»+%= 

54A+3A=  8%+%=  6V„+8»/„=  18"/«+"/i.= 

II.  One  piece  of  cloth  contains  %  of  a  yard,  and  another  2/3  of 
a  yard.     How  many  yards  in  the  two  pieces  ? 

Oral  Solution. — 3/4  is  equal  to  9/12,  2/3  is  equal  to  8/12,  9/i2  and  8/12  together 
are  equal  to  17/i2.  17/i2  =  l5/i2>  hence,  3/4  and  2/3  of  a  yard  of  cloth 
equal  15/12  yd. 

Illustration. — 3  fourths  and  2  thirds  of  a  sheet  of  paper  make  neither  5  fourths 
nor  5  thirds,  but  subdividing  both  into  twelfths  we  find  that  they  are  together  equal 
to  17/12  or  l5/i2. 

14-2.  Fractions  to  be  added  together  must  have  a  common 
denominator. 

Find  the  sum  of 

i2.  v2+y3=      i3.  %+«/,=      i4.  v3+7o=      i5. 76+7,2= 
V«+7t=  7*+78=  7+7,=   ...■      7s+7*= 

7+76=  7s+7«=  */*+%=  %+7.4= 

%+%=  V3+7,=  %+</,*  73+7.5= 

%+V»=  %+'/.=  */»+7«=  %+*/."=" 

i«.  Vt+Vi+7.  =  is-  7+7+74  +76  =  2o.  yt+7,+v4+v,+7,= 
»•  73+7+7.*=  is-  Vr+%+7i.+7«=  21. 7,+%+'%,= 


144 


STANDARD  ARITHMETIC. 


WRITTEN    WO  R  K. 

(43.  Example. — l.  Find  the  sum  of  7/18  and  8/15. 

Solution  with  Sheets  of  Paper. — By  subdividing  18ths  and  15ths  each  into  2, 
3,  etc.,  equal  parts, 

The  18ths  become  successively  36ths,  54ths,  72ds,  90ths,  108ths,  etc. 

The  15ths  become  successively  30ths,  45ths,  60ths,  75ths,  90ths,  etc. 

We  thus  find  that  18ths  and  15ths  can  both  be  changed  to  90ths  by  dividing 
each  18th  into  5,  and  each  15th  into  6  equal  parts;  and  hence  that  the  common  de- 
nominator of  the  equivalent  fractions  must  be  90,  which  is  the  1.  c.  m.  of  18  and  15. 

The  arithmetical  process  of  finding  the  sum  of  two  or  more  fractions  having 
unlike  denominators  embraces  corresponding  steps,  viz.,  Jirst,  finding  the  1.  c.  m.  of 
the  denominators ;  second,  reducing  the  given  fractions  to  a  common  denominator ; 
and,  third,  adding  together  the  numerators.     Thus, 

Explanation. — Having  found 
The  Arithmetical  Process.  the  1.  c.  m.  we  divide  it  by  18  and 

15,  and  thus  find  that  we  must 
multiply  the  7  (eighteenths)  by 
5,  and  the  8  (fifteenths)  by  6  to 
change  them  to  90ths.  Having 
performed  these  multiplications 
we  add  35/90  and  48/90  to  obtain 
the  sum  83/go' 

By  dispensing  with  such  part  of  the  written 
work  as  can  be  performed  without  the  aid  of 
the  pencil,  it  may  be  abbreviated  as  follows  : 

2.  Find  the  sum  of  %  %,  %  %,  % 

Explanation. — 24  being  the  least  common  multiple  of 
the  denominators,  the  given  fractions  may  all  be  reduced  to 
24ths.  •  In  '/4  there  are  6/24,  and  in  3/4  there  are  3  times 
6  or  18/24,  etc.,  etc. 


18=2X3X3 

90 

15=5X3 

18 

5X7=35 

15 

6X8=48 

Least  Com.  Mult. 

83/ 

h 

2X3X3X5=50 

24ths 

% 

18 

Vs 

3 

Ye 

20 

Vi. 

2 

% 

16 

9/     Oil/ 

/24— *      /S 


144.  little.— 1.  Reduce  the  fractions,  if  necessary,  to  fractions 
having  a  common  denominator. 

2.  Add  the  numerators,  and  under  the  sum  write  the  common 
denominator. 

3.  In  adding  mixed  numbers,  add  first  the  fractions,  then  the 
integers,  and  unite  the  results. 

Cantion. — If  any  arithmetical  process  results  in  a  fraction,  the  work  is  not 
complete  unless  the  fraction  is  expressed  in  its  lowest  terms.  Improper  fractions 
must  be  reduced  to  whole  or  mixed  numbers. 


FRACTIONS.  145 

Add  by  columns,  then  by  lines.     Test  results.*    (See  note,  p.  32.) 

3.       4.         5.        6.        7.        8.  13.    14.    15.       16.      17.    18. 

*  %+%  +  %  +%+%+%*        «.  Va+'A+V.  +%+%+% 
10.  %+%  +  Vu+VH-  %  +%6        20.  %+%+%  +  %  +%+% 

11.    %+%  +1Vl2+%+1V24+V5  21     %+•/,+•/„+  5/9  +l/i+l/f 

12.  y,+v»-f  %*+%+  %  +v,       22.  %+%+%  +  Vn+'A+v. 

23.  247/16+18%+503/10+18  %,+  4  %  +  2  %  +59%. 

24.  3%+  5%+17%  +  4  %2+371%4+171%2+70%. 

25.  157/9+243/4+38%+2713A6+331%5+19  %H$»£ 

26.  27.  28.  29.  30.  31. 

32.  4  %  +     3  %  +  4  %  +13  %  +17  VfU^ 

33.  63  %  +  21  %  +45  %0  +25  %  +142%0+22%2 

34.  227  7io+243*/4  +26  %  +36  %  +ll"/15+463/8 

35.  4383%0+65743/50  +382%0  +491%6+191%2+39% 

36.  840  %  +7602yioo+149yi0O+51  %  +15  %  +10% 

37.  Add  %+%+  %  +%+%  +  7i6+  %  +8A«+Vi. 

38.    Add  B/4+%+ll/lf+3/g+4/iB+  5/9  +  5/?  +4/l5+Vl8 

39.  Add  %+%+  %  +%+%2+%+i%0+%8+%4 


Applications. — 1.  Four  barrels  of  cider  contain  severally  25  % 
gal.,  237/12  gal.,  29%  gal.,  285/6  gal.     How  many  gallons  in  all  ? 

2.  What  is  the  total  weight  of  6  bales,  weighing*respectively 
5%  cwt.,  4%0  cwt.,  63/5  cwt.,  47/8  cwt.,  619/20  cwt.,  6%  cwt.? 

3.  Mr.  Abel  has  647/20  acres  in  farm-land,  355/8  in  meadow- 
land,  3219/50  in  woodland.     How  many  acres  in  all  ? 

4.  Of  5  brothers  the  youngest  is  11%  years  old,  the  second 
3y4  years  older,  the  third  2y6  years  older  than  the  second,  the 
fourth  2y5  years  older  than  the  third,  the  fifth  27/9  years  older 
than  the  fourth.     How  old  is  each  one  ? 

*  Any  two  or  more  columns  may  be  assigned  for  an  exercise.  Since  these  ex- 
amples are  self-testing,  no  answers  are  given. 


146  STANDARD  ARITHMETIC. 


Subtraction  of  Common  Fractions. 

ORAL 

EXERCISES. 

1.  If  there 

are  5/7  of  a  pie 

on  a  plate,  and  Harry  takes  2/i*  what 

part  of  the  pie 

remains  ? 

2. 

3. 

4. 

5. 

6. 

%    -  %    = 

%-%= 

5-%= 

'/-%= 

37s  -78  = 

17/      11/      

/18  —    /18  — 

V7-2A= 

6-'A= 

8-%= 

47io-71»= 

21/      12/      — 

/«  ""     /25  — 

%-%= 

3-%= 

»-7.= 

6V.  "%  = 

81/      __  25/      __ 
/loo —    /lOO — 

y6-7c= 

4-%= 

10-7r= 

7%  -'/,  = 

7. 

8. 

9. 

10. 

2%  -  %  = 

3  /io —    /lO111 

874- 

-474  - 

7 

V.  -3  V*  = 

45/13-10/i2= 

♦    /20          /20=:: 

59%  - 

-475  = 

33 

•A  -3  %  = 

6%  -  %  = 

a«/»-  %3= 

86?/,- 

-17s  = 

33 

3/    1  15/     

/16 — J-     /lC  — 

55/17-7n  = 

»%  -  'A  = 

987,i- 

-*/„= 

43 

4/           1  14/     _ 
/15 — *«    /15  — 

»4A  -  %  = 

3  /S7        Aj= 

147«- 

-37«= 

17: 

19/     —4.28/     __ 
/60 —  *     /60  — 

11.  Subtract  the  sum  of  7m>  744?  xVu»  27i4>  and  n{m  from  57m* 

12.  Subtract  the  sum  of  6/19,  3/10,  yi0,  "/w,  and  15/19,  from  7%. 

13.  Subtract  the  sum  of  13/100,  47ioo>  5^/100,  6n/ioo,  and  2y100, 
from  497ioo- 

14.  Sarah  has  76  of  a  yard  of  velvet.  How  much  will  she 
have  left  if  she  uses  74  of  a  yard  for  trimming  a  dress  ? 

Oral  Solution. — 5/6  is  equal  to  10/i2,  3U  1S  equal  to  9/12;  9/i2  being  sub- 
tracted from  10/]2  leaves  x/lf.  Hence,  Sarah  would  have  1/l2  of  a  yard  re- 
maining. 

145.  That  one  fraction  may  be  subtracted  from  another,  the 
two  must  have  a  common  denominator. 

1 4-6,  Mule, — 1.  Reduce  the  fractions,  if  necessary,  to  fractions 
having*  a  common  denominator. 

2.  Subtract  the  numerator  of  the  subtrahend  from  the  numerator 
of  the  minuend,  and  write  the  remainder  over  the  common  denomi- 
nator. 

3.  In  the  subtraction  of  mixed  numbers,  if  the  fraction  in  the 
subtrahend  is  greater  than  that  in  the  minuend,  take  1  unit  from 
the  whole  number  in  the  minuend  and  add  it  in  fractional  form  to 
the  fraction  of  the  minuend,  and  then  subtract. 


FRACTIONS.      "  147 


ORAL 

AND    SLATE    EXERCISES. 

Note. — The  oral  exercises  may  be  carried  as 

far  as  the  ability  of  the  pupU  will 

permit. 

15. 

16. 

17. 

18. 

id. 

%-%  = 

.%-%  = 

%  -  7s  = 

1%-%  = 

3%-%= 

Va-Vc  = 

7a-V.2= 

%  -  %= 

27a  -  %  = 

2%  -3A= 

%-%  = 

i/  _v  - 

/4         /24 — 

%  —  Vl2= 

4%  -'7.8= 

n  -%= 

/5 —  Ao=     . 

Vs— Vl6  = 

4/    5/     — 

/9  —     /18  — 

67s  -  7.6= 

9  /is— %  = 

%-%  = 

/5~~  /25  = 

%  ~~    716  = 

<    /l2 —    /24= 

?%*-%= 

73-%  = 

7,-%.= 

9/          11/     — 
'10 —     /20 — 

5   /ll —     /22  = 

67,.-%= 

/$—  lti= 

%  — %6  = 

%   —   %4  = 

8%5~   %C  = 

57,5-%= 

20. 

21. 

22. 

23. 

478-3716= 

?v«- 

-«%=           6% 

-1%  =         5 

%  -3%  = 

8%-l%'  = 

137s- 

-1%=         24% 

-2%  =         7 

/n— 3%2= 

8V.-1V8  =        25%-2%=        37%1-3%0=         9%  -5%7= 

24.  Subtract  $2%  from  $7%,  $6%,  $5«%,  $9%,  $4%. 

25.  Subtract  1%  lb.  from  6%6,  7%,  95/16,  ll7/^,  5%2  lb. 

26.  Subtract  3%  qt.  from  8%*,  9%8,  11%,  77/30  qt. 

27.  2%6-l3%8  =     34.     56%  -27%=      41.     48  % -7n/12= 

28.  3  %0-l17i2o=     35.  165  :%  -39%2=       42.  824  %  -6  %2= 

29.  74  %5-21%0  =     36.  283  %  -46%4=       43.  936  %-±%0= 

30.  68"/20-9  %   =     37.  394  %5-53%0=       44.  141  %0-3  »/tts= 

31.  1513/18-9  %   =     38.  443  %0-31%6=       45.  475  8/S6-7  %s= 

32.  6015/16-3  %   =     39.  527  %7-13%2=       46.  718  %2-8  %,= 

33.  251%6-2  7/12  =     40.  136%-  4%  =       47.  248 %-5  %  = 

48.  From  585%1+4561%2+3541%0  take  8%+0%+lYi* 

49.  Take  77/8  inches  from  165/6  in.,  from  187/23  in.,  from  21%7 
in.,  from  23 7/H  in.,  from  297/29  in. 

50.  Take  137/n  minutes  from  23%,  47  %9,  31%,  5617/23,  35  % 
minutes. 

61.  Subtract  each  number  from  the  next  to  the  right ;  add  the 
first  number  and  all  the  remainders  together.     (Sec  No.  l,  p.  96.) 
%  *V»  .2%,  3%,  3%,  4%,  5%6,  6%,  300. 


148  STANDARD  ARITHMETIC. 

Applications. — l.  A  train  reached  Chicago  at  10  o'clock ;  it 
had  made  the  trip  from  Milwaukee  in  3  %  hours.  At  what  time 
did  it  start  from  Milwaukee  ? 

2.  A  person  bought  a  piece  of  linen,  measuring  60%  yards. 
After  the  linen  was  washed  it  measured  only  59%  yd.  How 
much  had  it  shrunk  ? 

3.  A  grocer  had  2  cheeses,  one  weighing  38%  and  the  other 
4517/32  pounds.  He  sold  7%  lb.  of  each.  What  was  the  differ- 
ence between  the  weights  before  and  after  the  sale  ? 

4.  What  is  the  difference  in  age  of  two  persons,  now  73  7/12  and 
46  %  years  old  respectively  ?    What  will  it  be  10  years  hence  ? 

5.  A  thermometer  showed  at  noon  73  %>  degrees  above  zero.  At 
6  o'clock  r.  m.  it  showed  65  %  degrees.    How  much  had  it  fallen  ? 

6.  A  grocer  received  a  box  of  tea  weighing  247a  pounds.  The 
weight  of  the  box  alone  was  3  %6  lb.    How  much  did  the  tea  weigh  ? 

7.  From  $120  %  the  following  sums  were  taken  :  $6%,  $12%, 
$2617/20,  $20n/50.     What  was  the  remainder? 

8.  Last  fall  we  received  17  %  tons  of  hard  coal ;  in  the  spring 
1%  tons  were  left.     How  much  had  we  used  during  the  winter  ? 

9.  A  boy  said,  "If  I  had  $5%  more  than  I  have,  I  should 
have  $21%."    How  much  did  he  have  ? 

10.  A  flag-staff  483/4  feet  high  was  broken  off  near  the  top  by 
a  storm,  so  that  it  measured  only  41%  ft.  How  long  was  the 
piece  that  had  been  broken  off  ? 

11.  A  farmer,  owning  388%  acres  of  land,  bought  in  addition 
251%,  and  then  sold  parcels  containing,  respectively,  84%,  26%, 
38 %,  29n/12,  93%,  and  84%0  acres.     How  much  had  he  left  ? 

12.  A  grocer  cut  1%,  2%,  2%,  3%2,  1%,  5%,  9%,  1%, 
2%,  %6,  kz,  2%  pounds  from  a  cheese  which  weighed  437/8  lb. 
How  many  pounds  remained  ? 

13.  One  man  cuts  3%  cords  of  firewood  per  day,  another  3% 
cords  per  day.  The  first  works  7  days,  the  second  6  days.  How 
much  does  one  cut  more  than  the  other  ? 


FRA  CTIONS. 


149 


Multiplication  in  Common  Fractions. 
Example. — l.  Three  times  3/4  inch  are  how  many  inches  ? 
Solution.— 8/iX%  in.=9/4  in.  =274  in. 

2.  Three  times  23/4  inches  are  how  many  inches  ? 
Analysis.— 3X2%  in.=s/iX11/4  in.=8%  in.=8y4  in. 

3.  %  of  2  are  how  many  ? 
Analysis.-s/4of  */1=%=lV*. 

Note. — After  a  simple  fraction  the  sign 
x  should  be  read  "of"  not  "times." 

4.  3/4  of  3/4  are  how  many  ? 
Analysis—I/,  of   y_,/w<.    %  of  %=,/jt; 

«/4  Of   %=%,. 

5.  3/4  of  2%  are  how  many  ?    . 

Illustration.— The  line  of 
the  arrow  cuts  off  1/A  of  the 
2  3/4  squares  represented,  leav- 
ing  3/4   of    the   23/4    squares  "^ 


*-> 



> 

>;; 

above  it. 

Analysis.-^  of  2%=%  of  %  ;  %  of  %=% ;  %  of  «%»f»Ai ; 

%  of  "U=»/1%=2Vt.. 

6.  33/4x23/4  are  how  many  ? 

Illustration. — Copy  the  last  diagram  four  times,  omitting  the  arrow  in  each  line 
of  squares  except  the  fourth.  Thus,  above  the  line  of  the  arrow  3  3/4  times  23/4. 
squares  will  be  represented,  illustrating  the  following  analysis. 

Analysis.-33/4X2%=15/4  of  »/4  J  V*  of  %=%  J  %  of  %=UU  S 

15/4  0f    »/4=166/l«=106/l«. 

The  parts  of  the  several  analyses  printed  in  heavy  type  are 
the  only  parts  needed  in  a  written  solution.     Whence  the 

14-7.  Utile,— Reduce  integers  and  mixed  numbers  to  improper 
fractions.  Multiply  the  numerators  together  for  the  numerator  of 
the  product,  and  the  denominators  together  for  the  denominator  of 
the  product. 


150 


STANDARD  ARITHMETIC. 


For  practice  in  multiplication  of  fractions,  the  pupil  may  complete  the  follow- 
ing tables,  and  construct  similar  ones.  When  the  multiplier  is  a  fraction,  he  should 
substitute  the  word  "of"  for  "times"  in  all  oral  exercises. 


1 

2 

3 

4 

5 

6 

7 

1/ 
/% 

1 

IX 

2 

^ 

3 

3X 

A 

A 

1 

IX 

IX 

2 

1/ 
7l 

A 

3/ 

A 

1 

1/ 
/5 

/5 

3/ 
/6 

1/ 

v% 

a 

1 

X 

X 

tT 

1  */ 

75 

% 

6/      7/ 
/7      /7 

2 

1 

IX 

IX 

B< 

3 

J# 

2 

^X 

4 

2 

2% 

3 

i 

5 

2% 

6 

3 

7 

1 

1/ 
/2 

A 

7* 

X 

X 

1/ 
/7 

X 

1/ 
7± 

/6 

/8 

X, 

1/ 
/3 

X 

1/ 
/9 

A 

1/ 
/8 

A 

&c. 

1 

A 

A 

1/ 

/7 

X 

/f 

6/< 

X 

X 

As 

A* 

3/ 
735 

4/ 
/35 

X 

As 

±7 
/35 

A 

77 

A 

X 

X 

A 

A 

A 

1 

ORAL    EXERCISES. 
Note. — Let  the  oral  exercises  be  carried  as  far  as  possible. 


1. 

2. 

3. 

4. 

?xy2= 

7^X26  = 

6X%= 

VioX23 

5X%= 

V6iX72= 

7X%= 

"/15X6O 

4X%= 

V25X36= 

9X%= 

7/12X29 

6xy4= 

V38X47= 

3X%= 

Vs  X13 

8X%= 

V27X54= 

5X6A  = 

9/nXl5 

9xVr= 

V56X63= 

4X%= 

%  X25 

iox%= 

V2,X33  = 

8X%= 

%5X39 

FRACTIONS,  151 

Note. — In  written  work,  always  cancel  factors  that  are  common  to  both  nu- 
merator and  denominator. 

5.  How  much  is  6  X  %  day  ?  %  d.  ?    %  d.  ? 

6.  How  much  is  9x%  lb.?  %  lb. ?  17/16  lb. ? 

7.  How  much  is  17X*1/*  ?  $7/io  ?   *SA  ? 

8.  What  is  %  of  1  hour  ?   6  h.  ?     9  h.  ? 

9.  What  is  7«  of  3,  5,  7  feet? 


10. 

11. 

12. 

13. 

V,  of  %= 

%  X%= 

"A  of  %= 

%iX   17= 

V,  of  •/,= 

/igX  /«= 

% "  %= 

19x13/25= 

V.  of  %= 

%  xy5= 

4/     <<    3/  

/9            /8— . 

8/„X  28= 

'A  of  y4= 

A2X   /6  = 

2/     "    5/  — 
/3            /6  — 

51X14/29= 

%  of  •/«= 

3A  xVs= 

Ve  "  %= 

/63X   35= 

%  of  %= 

Ve  X%= 

Vi  "  %= 

46  X33^- 

*A  of  y,= 

7s  xv«= 

4/     <<    5/  _ 

%X   29= 

14. 

15. 

16. 

17. 

3%X5= 

65/lsX8= 

13/3x1V6= 

uAx"A« 

4%X6= 

7?/8X9= 

19/5X19/8  = 

MAx,3/9= 

5%X7= 

9%  X8= 

%x%= 

7.x»A= 

18.  What  is  Vb  of  $%  ?   %  of  $  V10  ?   %  of  I  %  ?   Vs  of  *%  ? 

19.  What  is  %  of  %  hour  ?   %  of  %  h.  ?  %  of  %  h.  ? 

20.  What  is  y16  of  %  lb.  ?  yi6  of  %  lb.  ?  Vie  of  %  lb.  ? 

21.  What  is  %  of  %  ft.  ?  %  of  %  ft.  ?  %  of  %  ft.  ? 

22.  What  is  %  of  %  week  ?  %  of  %  wk.?  %  of  %  wk.  ? 

23.  What  is  %  of  %  qt.?   %  of  %  qt.?   %  of  9/10  qt.  ? 

24.  Multiply  %  by  1%  ;  %  by  2%  ;   %  by  3%  ;  6/n  by  44/7. 
26.  3  %  X  %=         26.  %  X  7  %=        27.     8  %  X  %=        28.  %  X  6  %= 

224xy5=  %X3%=  7y4X%=  %X2%= 

42/5xy6=  y3X5y2=  54%X3A=  4A  X8%= 

33Axy8=  y5X42/5=  253/8X%=  VnX7V7= 

29.  Multiply  3%,  7%,  9%,  6%,  8%,  5%,  4%,  each  by  %, 

30.  What  is  %  of  3n/12  ?  of  5%i  ?   of  6%  ?  of  102/3  ?  of  9%  ? 


152 


STANDARD  ARITHMETIC. 


(4-8»  Since  mixed  numbers  may  be  reduced  to  improper  fractions,  and  inte- 
gers may  be  expressed  in  fractional  form,  the  general  rule  may  be  applied  to  all 
cases  of  multiplication  in  which  fractions  are  involved,  but  when  the  numbers  arc 
large,  the  method  is  awkward,  and  requires  more  figures  than  the  following  pro- 
cesses.    In  business  calculations,  the  rule  is  seldom  followed. 


Example.— l.  Multiply  85  by  17%. 

85 

1'  k     Analysis. 
3)170         =2x85 


5fj%    =  2/3x85 
595 
85 


(- 


x85 


1501%    =  172/3x85 

3.  Multiply  645%  by  328  %. 


645% 

328% 


Analysis, 


4)1935 
3)  656 


483% 


%  =3Ax2/3 
=  3/4x645 
r=328x2/3 


=328x645 


212262  %    =  3283/4x6452/. 


2.  Multiply  58%  by  29. 
58% 

jf         Analysis. 
4)87         =29x3 
21%    =29x3/4 

522 
116 

1703%    =29x583/4 


\- 


29x58 


Explanation. — Beginning  at 
the  right,  as  in  multiplication  of 
integers,  we  multiply  separately 
the  fraction  and  integer  of  the 
multiplicand,  Jlrst  by  the  fraction 
and  second  by  the  integer  of  the 
multiplier. 

The  work  at  the  left  indicates 
the  steps  by  which  we  obtain  the 
product  of  the  integers  and  frac- 
tions. 


Multiply 
1.  8%  by  12 
15%  by  16 

7%  by    9 
6%  by  14 

5.  2%  by  3 
4%  by  5 

7%  by  6 


ORAL    EXERCISES, 


13%  by 

17%  by 
18%  by 


15%  by  12 


6.  4%  by  5 

6%  by  8 

8%2by7 


3.  6  by  2% 
12  by  3% 
14  by  6% 

16  by  4% 


7.  9  by  3% 

13  by  6% 

18  by  3% 


4.  12  by  7% 
18  by  4% 
21  by  6% 

15  by  7% 

8.  37  by  2% 

16  by  6% 

27  by  2% 


FRACTIONS. 


15S 


SLATE    EXERCISES 


8y5x6%= 

*%X3%= 

5%X2%= 
7%X3%=  . 

5. 

6. 

7. 

8. 


6%X4%= 
44%X5%= 
82%X7%= 
27%  X  3%= 


1     /25X3    %3  — 

6^X9%= 
9,8/i9X7%= 


3. 

14%Xll%0= 

i3%x25%  = 

16%Xl3%  = 

24%X28%  = 
Find  the  product  of  2%X$30%0 ;  1%X$25%  ;  47/8X*19%. 
Multiply  $18%  by  153/7 ;  $2713/21  by  3817/20 ;  41%  by  20%. 
2%5X53/4X16%=?    %  of  %  of  %  of  y21  of  %=? 
What  number  is  3%  times  35%  ?    5%  times  63/8  ? 

9.  There  are  4y8  lb.  in  a  package.     How  many  pounds  in  8y4 
such  packages  ? 

10.  Eeduce  2/3X3/4X4/5X5/9X9/32X1%7  to  a  simple  expression. 


Applications. — Note. — In  business  it  is  customary  to  drop  a  fraction  in  the 
result,  if  less  than  l/t$,  and  to  add  \f  to  the  integer,  if  the  fraction  is  equal  to 
or  greater  than  J/20-  The  pupil  should  here  be  required  to  obtain  the  exact  an- 
swer, and  to  write  the  result  in  business  form  under  it. 


11-38.  What  is  the  cost  of 
37y2  bushels  of  potatoes  at  75%£  ? 
345  yards  of  cloth  at  90  %£  ? 
17%  feet  of  oilcloth  at  3%^  ? 
43  y2  quarts  of  cider  at  5%jfi  ? 
387  bushels  of  oats  at  43%$*  ? 
40%6  pounds  of  starch  at  18*/*^  ? 
345  barrels  of  apples  at  $32/5  ? 
47  y2  sacks  of  flour  at  $23/5  ? 
53  y2  pounds  of  cheese  at  93/4^  ? 
17  sacks  of  rice  at  $143/5  ? 
64  pecks  of  beans  at  173/5^  ? 
27  4/5  pecks  of  potatoes  at  23  y8#  ? 
328 3/4  pounds  of  butter  at  433/rf  ? 
179/j6  pounds  of  bacon  at  12  y2^  ? 


56y2  pounds  of  tea  at  $3/4  ? 
63/4tonsof  coal  at  $33/4? 
15%  yards  of  ribbon  at  37%^  ? 
24%  gallons  of  oil  at  85^  ? 
3  quarts  of  nuts  at  12  y8# 
300  bushels  of  rye  at  94y80  ? 
12  y2  yards  of  lace  at  $5%  ? 
173/4  pounds  of  honey  at  18%$*  ? 
4y2  barrels  of  herring  at  $3  y4  ? 
325  pounds  of  beef  at  11  %£  ? 
63  bl.  of  cranberries  at  $12  y8  ? 
173/8  bu.  of  strawberries  at  $45/8? 
37%  yards  of  velvet  at  $4%  ? 
48  3/4  yards  of  carpet  at  $l3/4  ? 


154  STANDARD  ARITHMETIC. 

39.  How  many  square  feet  in  a  square,  each  side  of  which 
measures  3  %  feet'?  3  %  inches?  3%  yards? 

Draw  a  square,  each  side  measuring  2  3/4  inches.  Divide  each  side  into  fourths 
of  an  inch,  and  draw  lines,  cutting  the  square  into  small  ones,  each  a  fourth  of  an 
inch  square.     How  many  of  these  small  squares  ?     How  many  square  inches  ? 

40.  Find  the  number  of  square  inches  in  a  rectangle  5%  ft. 
long  and  7%  inches  wide. 

41.  What  is  the  cost  of  9  tons  of  coal  at  $3  %,  with  cartage  at 
$y4  per  ton? 

42.  If  the  salary  of  an  officer  is  $1700,  how  much  does  he 
receive  in  %  %  %,  %  year  ?  How  much  in  %,  3/4,  4/5,  */„ 
year  ? 

43.  A  laborer  earns  $20%  a  week.  How  much  in  %  year  ? 
In  1%  yr.  ?     (Count  52  weeks.) 

44.  My  neighbor  pays  $700  rent  per  year.  How  much  is  that 
per  month  ? 

45.  I  buy  some  lots  containing  respectively  %,  %,  13/25,  4:/50, 
63/10,  1913/2o  acres.    What  is  the  cost  of  the  whole,  at  $48  per  acre  ? 

46.  How  many  square  feet  in  the  surface  of  a  stone  slab  27/s 
feet  wide  and  4%  feet  long? 

47.  Four  boxes  of  hardware,  weighing  respectively  3y4  cwt., 
4%  cwt.,  4y8  cwt.,  and  43/10  cwt.,  cost  16^  freight  per  cwt.  What 
is  the  freight  on  each  box,  and  on  the  4  boxes  ? 

Find  the  sum  to  be  paid  for 
48.  5y2  lb.  sugar,  at  9%0  49.  18  yd.  calico,  at  9^ 

63/4  lb.  coffee,  at  32  \j4  20  %  yd.  alpaca,  at  42  %^ 

24/5  lb.  rice,  at  84/^  19%  yd.  shirting,  at  170 

143/8  lb.  flour,  at  4%^  25  yd.  ribbon,  at  W*fc 

33/4  lb.  butter,  at  23  %#  10  doz.  buttons,  at  $% 

2%  lb.  cheese,  at  11  %#  3  cloaks,  at  $18% 

3%  doz.  eggs,  at  20^  10  yd.  velvet,  at  $33/4 

2%  lb.  starch,  at  12  %^  jtt%  yd.  velveteen,  at  $1  % 


FRACTIONS. 


155 


Division  in  Fractions. 

Note. — In  the  first  two  exercises  the  square  is  the  unit.     In  the  third  exercise 
the  linear  inch  is  the  unit. 

1.  a.  How  many  times  1  in  3,  2%,  2,  1%  ? 
What  part  of  1  is  contained  in  %  ?  What 
part  of  2  in  */,?    In  l1/,  ?  etc. 


b.  How  many  times  %  in  1  ?    In  2  ?    In  3  ?     How  many 
times  1%  in  3  ?   2%  in  3  ?    (5/2  *«  6/2?) 

e.  How  many  times  %  in  l1/,  ?    In  21/,  ?    I1/,  in  2%  ? 


2.  «.  How  many  times  1  in  1%?    In  2y3?    What  part  of  1 
is  in  %  ?    What  part  of  2  ?    What  part  of  2     . 
is  in  %!     In  iy3? 

#.  How  many  times  y3  in  1  ?   In  2  ?   How 


many  times  %  in  2  ?  In  3  ?    How  many  times  1  %  in  3  ?  2  y3  in  3  ? 
c.  How  many  times  y3  in  5/3  ?  In  2%  ?     How  many  times  iy3 

in  2%  ?   %  in  22/3  ?   1%  in  3  ?   1%  in  2%  ? 


Inches       / 

k                                        > 

V                                                                          / 

A 

Halves 

j 

\ 

/ 

V 

J 

i 

Thirds 

y 

k 

/ 

^ 

y 

k 

J 

v 

y 

k 

I 

k. 

Fourths 

) 

\ 

y 

k 

j 

\ 

y 

V 

/ 

k 

y 

k 

y 

k 

/ 

k 

I 

V 

Sixths 
Twelfths 

i 

k 

/ 

V      i 

k  y 

\ 

> 

k 

y 

V       / 

k   / 

k 

/ 

k. 

y 

k 

/ 

k     / 

k     7 

k 

J 

V 

12 


Note. — The  points  of  arrows  divide  the  inches  into  halves,  thirds,  etc.  By 
following  the  shafts  downward  equivalents  in  smaller  parts  are  found. 

3.  a.  How  many  times  2  in  23/4  ?    What  part  of  3  in  %,  1%  ? 

I,  How  many  times  y6  in  2  ?  In  2y8?  In  3  ?  How  many 
times  %  in  1  ?  In  7/12  ?  In  2  ?  How  many  times  %  in  3  ? 
What  part  of  2  is  %  %  ?    What  part  of  3  is  1%  %  %  ? 

c.  How  many  times  %  in  %  ?  5/12  in  %  ?  %  in  3/4  ?  %  in  %  ? 
What  part  of  %  is  in  %,  %  ?     What  part  of  %  in  %  ?  etc. 


156 


STANDARD  ARITHMETIC. 


^ 

I 

— 

— 

— 

— 

— 

— 

t 

J 

) 

4.  Divide  %  by  % 

Illustration. — To  find  how  many 
times  3/5  is  contained  in  3/4,  the 
5ths  and  4ths  are  subdivided  into 
parts  of  the  same  size  by  dividing 
each  5th  into  four  and  each  4th 
into  five  equal  parts,  thus  making 
12/20  and  15/2oj  whence  we  see 
that  3/5  is  contained  in  3/4  as 
many  times  as  12  is  contained  in 
15  =  15/12  =  174. 

149.  The  arithmetical  solution  is  performed  on  the  same  prin- 
ciple as  the  solution  by  diagram,  the  folding  of  paper,  etc.,  thus  : 

Indicating  the  division  by  writing  the 
divisor  under  the  dividend  (see  Art.  72,  §  3), 
we  reduce  both  fractions  to  20ths,  20  being 
the  least  common  multiple  of  4  and  5,  and 
divide  the  numerator  of  the  dividend  by  the 
numerator  of  the  divisor. 

But,  since  the  common  denominator  does 
not  affect  the  quotient  (16/i2)>  the  work, 
printed  in  italics,  by  which  it  is  obtained 
may  be  omitted.  The  written  process  would 
then  appear  as  at  the  right. 


3/X5. 


15/ 

— f». 


15/ 


3/  X4_12/ 
/5Xf~    1 20 


3/X5/: 

4X/3: 


45/ 


in 


=iV« 


=i% 


150.  Mule. — Invert  the  divisor  and  multiply  the  numerators 
together  for  the  numerator  of  the  quotient,  and  the  denominators 
for  the  denominator  of  the  quotient. 

Note. — Since  integers  and  mixe'd  numbers  may  be  expressed  in  the  form  of  im- 
proper  fractions,  this  rule  applies  to  ail  cases  of  Division  in  Common  Fractions. 

151.  The  following  analysis  leads  to  an  equivalent  rule  : 

3/5  is  contained  in  1,  or  5/6,  five  thirds  times,  and  in  3/4  it  is  contained  3/4  of 
e/q  times;  hence  we  have 


3/y5/. 
•  /4X  /8" 


.15/ 


;=iy* 


3/^3/ 
/4  •  /5~  /4"  /3-    l\%; 

152.  An  inverted  fraction  shows  how  many  times  the  fraction 
is  contained  in  one,  and  is  called  the  reciprocal  of  the  fraction. 
Hence,  for  dividing  one  fraction  by  another,  we  have  the 

liule.— Multiply  the  reciprocal  of  the  divisor  by  the  dividend. 


FRACTIONS. 


157 


153.  In  dividing  a  fraction  by  an  integer,  a  part  of  the  written 
work  required  by  the  rule  may  be  omitted,  as  follows  : 

Second,  when  the  numerator  of 
the  fraction  is  not  divisible  by  the 
integer,  as  in 

Example.— 2.  Divide  5/7  by  3. 
(What  part  of  3  in  5/7  ?) 

The,  process  of  division,  under  the 
rule,  would  bo 

/7  *  6~  /3X  jt~  l%V 

but  we  can  multiply  the  denominator 
directly  by  3  without  writing  out  the 
second  step ;  hence,  we  need  to  put 
down  only 

1 54-.  Hence,  dividing  the  numerator  or  multiplying  the  de- 
nominator of  a  fraction  divides  the  value  of  the  fraction. 

155.  In  dividing  a  mixed  number  by  an  integer,  by  a  fraction, 
or  by  a  mixed  number,  the  written  work  may  be  as  follows  : 

Example.— 3.  Divide  379%  by  6. 

Explanation. — Six  is  contained  in  3793/4  63  times,  with  a  re- 
6)379%  mainderof  l3/4. 

637/  l3/4  =  7/4;    7/4-J-6=7/24,  which,  being  annexed  to  63,  gives 

the  entire  quotient,  637/24. 

Note. — In  the  two  following  examples  we  multiply  both  divisor  and  dividend 
by  the  denominator  of  the  divisor  in  order  to  get  rid  of  the  fraction  in  the  divisor. 
This  makes  the  division  more  convenient  and  does  not  alter  the  value  of  the  quotient. 
The  process  then  becomes  the  same  as  in  the  preceding  solution. 


First,  when  the  numerator  of 
the  fraction  is  divisible  by  the  in- 
teger, as  in 

Example.— 1.  Divide  «/7  by  3. 
(What  part  of  3  in  6/7  ?) 

The  process  of  .division,  according 
to  the  rule,  would  be 

2 

/7-^--/$x/7-/7? 

but  this  is  equivalent  to  dividing  the 
numerator  directly  by  the  integer,  thus, 

6/-3 

If6' 


4.  Divide  379  %  by  %. 

%)  379% 

3 3_ 

6)1137% 

568% 


5.  Divide  349%  by  2%. 
2%)349% 


9)  1398 % 


155  % 


158  STANDARD  ARITHMETIC. 


Divide 

ORAL    EXERCISES. 

1. 

2. 

3. 

4. 

V4  by  V. 

V.  by  Vn 

%  by  •/» 

V,  by  V,6 

%  by  % 

Ve  by  V» 

%  by  Vi, 

'A  by  % 

%  by  %i 

%  by  >/M 

%  by  >/tt 

7t  by  V33 

%  by  % 

'A  by  % 

7s  by  % 

V.  by  »/„ 

Question. — In  which  of  the  above  columns  do  the  quotients  increase  as  you 
descend  ?     Why  do  they  increase  ?     In  which  do  they  decrease  ?     Why  ? 

6.  Divide  %  by  %  ;  %  by  %  ;  %   by  %  ;  %   by  »/„ ;  %  by  •/„. 

6.  »      %  by  >/,  ;  %  by  %  ;  %   by  %;  %  by  %,;  %  by.'/, . 

7.  "      %  by  %,;  %  by  '/„;  %  by  %  ;  '/„  by  %  ;  %  by  %,. 
8.  How  many  times  may  ys  of  a  pound  be  taken  from  6/7  lb., 

V,  lb.,  %ib.,  Vioib.? 


9. 

10. 

ll. 

12. 

%  +4= 

V»+2= 

%  +  4= 

%-«-? 

7.  -3= 

7.3-9  = 

%+»*? 

'7,7-3 

Ve  -7= 

%  -2= 

7,1-  v= 

%*4 

'/n+3= 

%  -3= 

«/,5-16= 

%  +8, 

%  -6= 

'713-4  = 

7,  *  8= 

%  -8: 

13.  Divide  6%  by  4% ;  5%  by  2% ;  38/n  by  iyi3 ;  2%  by  % 

14.  Divide  9%0  by  4y7 ;  2%  by  9%0 ;  7%  by  2%  ;  6%  by  3%. 
15.  23/4-r-7=  16.  56/7-f-9  =  17.  94/5-8=         18.  8%-4-5= 

6%-$-8=  3%-*-7=  8%-r-8=  3%-5= 

9%-S-5=  7%+$=  7%+6=  4%-s-8= 

19.  Divide  5  by  % ;   7  by  %  ;   6  by  %;   12  by  % 
20.  18+%=  21.  14-4-%  =  22.  39-^%!=  23.  27^%3= 

36-%=             41-yi9=              23-5/13=  2o-y10= 

84-h%=               35-%=                49-%  =  19+%,= 

17+%=               71+%=               21+%=  23-%7= 

24.  Divide  23  by  4%;   17  by  5%;   48  by  3%;  59  by  7%. 

25.  Divide  25  by  3%;   24  by  7%;   39  by  82/3;  72  by  2%. 


FR ACTIONS. 

159 

ORAL    AND 

SLATE     EXERCISES. 

Note.— Let  the  oral .  exercises 

be  carried  as  i 

;ar  as  time  and 

the  ability  of  the 

pupils  will  permit. 

1.  1-4-6=         2 

.  4^3=        3 

4+%= 

4.  5+%= 

5.  S-»-l%= 

1*8= 

7-5-4= 

3+yt= 

6-7,= 

5*-8%= 

2-^-3  = 

6-r-5  = 

8+V,= 

8-=-%= 

27-=- 7%= 

2-f-4= 

14^8= 

5+%= 

4+%= 

15-3%= 

5-4-7= 

8^-5  = 

7+%= 

10+%= 

39+9%= 

6.  %-*-4= 

7.  %  +y;  = 

8.  2% 

+%  =           9 

.  15%+4%= 

%+$= 

%  +■/•  = 

3% 

+%  = 

17%-h3%= 

Ve-3  = 

%  -s/t  = 

3% 

+7n=  ' 

16%-=-5%= 

%-6  = 

14/       .   7/     — 
/15   •     /15 — 

5% 

+%> 

297,-9%= 

10.  How  many  times  $4%  in  $26%  ?     In  $37%  ?     In  $45%? 

11.  3%  hours  is  what  part  of  19%,  272/7,  38%  hours  ? 

12.  What  part  of  18 %6,  24%,  30%,  49 %  lb.  is  6%6  lb.? 


13.  %-%  = 

74-78   = 

1/  .  1/  _ 

h~  /36  — 

A—  ho— 
17.  %+?/ 


14. 


10/  _^2/  — 
/21—  /3  — 

12/  .  4/  — 
/24—  /6  — 


15. 


18.  % 


is.%-%; 


%; 


20/     _i_5/     _ 

/35  •  /7  — 
35/    _i_5/     — 

/42  •  /6  — 
18/    _i_6/     _ 

/33~  /ll  — 

/72~=~  /18  >        /75"^"  /15  > 

/§"*"  A  j        A-!-  As >        A8"5"  A  5        A"*"  A 

3/     •   5/    .       3/  _;_2/    .       2/  j_4/    .       3/    *  5/ 
/8   •     /6  )         /5   •     /5  >         /7—  /9  5         /7   •     /12' 


A—  As— 

16-   %  +%5= 

/«**- A*33 

Ai"^-  A3— 

%+y.  = 

25/       •   5/     

/36~  /72  — 

A—  A5= 

A  t"  As1^ 

A—  As= 

/l8"*"7M  = 

/«"*"  Ae  5 

5/     _i_5/ 
/36  —  /9« 

20.  IV3-5-V,  =   21.  ll'/3+l%6=  22.  3%  -=-%  =  23.  5V,2-4-4%  = 

3%-=-%2=  »%+»%  =  2Vu-%=  3% +2%  = 

n+5/w=  45%H-32/2s=  1%+%=  5%0-M%  = 

4%+%5=  25V2+4%=  l%+%3=  6%-=-3%6= 

!%+%«=  27%-=-2%=  2V15+%,=  7% +4%,- 

24.  Divide  %  by  2%;    %  by  3%;   %  by  1%;   %  by  2%  5 
%  by  3%. 

25.  Divide  %  by  7%;   %  by  3% 
%  by  3%. 


%  by  4%;   %  by  5%; 


160  STANDARD  ARITHMETIC. 

Applications. — l.  If  4  yards  of  ribbon  cost  $%,  what  will  1 
yard  cost  ? ' 

Analysis.— If  4  yards  cost  $5/?,  one  yard  will  cost  1/4t  of  $5/7  =  $5/2  8- 

2.  A  farmer  sold  5  dozen  eggs  for  §u/&.  How  much  was  that 
per  dozen  ? 

3.  In  six  days  a  man  plows  5/14  of  a  field.  At  this  rate,  how 
mnch  does  he  plow  in  1  day  ? 

4.  If  a  weaver  earns  $93/20  per  week  (6  days),  what  does  she 
earn  per  day  ? 

5.  If  %  lb.  of  coffee  cost  $3/8,  what  will  1  lb.  cost  ? 

Analysis.— If  4/s  lb.  cost  $3/8,"  V#  will  cost  l/A  x$3/8  =  $3/32,  and  5/6  lb. 
will  cost  5x3/32  =  $15/32. 

6.  If  a  traveler  can  make  %  of  his  journey  in  3/7  of  a  month, 
what  time  will  the  entire  journey  require  ? 

7.  If  %  of  a  bar  of  gold  weighs  5/12  lb.,  what  is  the  weight 
of  the  bar  ? 

8.  How  many  are  %  of  2%  dozen  ?    %  of  2%  gross  ? 

9.  A  garden,  containing  148 4/5  □  yd.,  is  to  be  divided  up  into 
beds  of  12%  □   yd.  each.     How  many  such  beds  will  there  be  ? 

10.  A  quantity  of  grain,  weighing  110 */4  cwt.,  is  to  be  put  into 
bags,  each  holding  1%  cwt.     How  many  bags  are  required  ? 

11.  How  many  yards  of  cloth  at  $3/8  a  yard  can  be  bought  for 
$2,  $5,  $7,  $9,  $4,  $23  ? 

12.  How  many  bushels  of  potatoes  at  $2%5  per  bu.  can  be 
bought  for  $6,  $8,  $11,  $13  ? 

13.  How  many  times  may  l8/9  quarts  be  drawn  from  a  can 
holding  17  quarts  ?    From  one  holding  22%  quarts  ? 

14.  If  a  laborer  can  mow  a  field  in  714/i9  da}^s,  how  much  of  it 
can  he  mow  in  1  day  ? 

15.  Divide  $22,500  among  the  7  members  of  a  family  so  that 
each  of  the  4  older  ones  may  receive  a  third  more  than  one  of 
the  younger. 


FRACTIONS.  161 

Complex  Fractions. 

If  a  slip  of  paper,  a  melon,  or  other  object,  is  cut  into  3 
equal  parts,  one  and  a  half  of  these  parts  are  1%  thirds,  2%  parts 
are  2V8  thirds,  etc. 

Show  by  the  use  of  objects  what  is  meant  by  — 3,  -~,  etc. 

0  o 

156.  A  complex  fraction  is  a  fraction  having  a  fraction  or 
mixed  number  for  its  numerator  and  an  integer  for  its  denom- 
inator. 

Reducing  Complex  to  Simple  Fractions.    Example. — l.  Reduce 

21/ 

-£*  to  a  simple  fraction. 

Analysis. — If  each  fourth  of  an  object  be  divided  Written  Work, 

into  3  equal  parts,  4  fourths,  or  the  whole,  will  con-  %!/      21Ax3  =  7/ 

tain  4  times  3,  or  12,  and  21  /3  will  contain  2l/3  times  —r-  =~7~  yQ /l  2 

3,  or  V,  of  them,  hence  2l/3  fourths  are  equal  to  7/12.  ' 

Illustration. — The  mode  of  demonstrating  the  foregoing  analysis  by  means  of 
objects  is  sufficiently  indicated  by  the  analysis. 


Reduce  to  simple  fractions : 

.   2%             3V, 
U                 8 

3.^ 

7 

.    5% 

■    2% 

6'T 

«•? 

7    %             8    ^ 
7-   7            8-    8 

11 

3/ 

10.  A 
15 

n.  ^ 

12 

»a 

2356'/, 
13,  ~3872~ 

14. 

483  Vn 

"^286 

15. 

% 

989 

157.  Expressions  in  which  fractions  or  mixed  numbers  occur 

4 
as  denominators,  as  —p-,  are  not  properly  fractions,  though  they 

5   /2 

are  commonly  classified  as  such.  They  only  indicate  division  (see 
Art.  72),  and  are  reduced  by  performing  the  division  indicated, 
or  by  multiplying  divisor  and  dividend  of  the  complex  expression 
by  the  least  common  multiple  of  the  denominators  of  their  frac- 
tional parts  (see  Art.  85). 


162  STANDARD  ARITHMETIC. 

SLAT  E     EXERCISES. 

Reduce  to  simple  fi  actions  : 

,:  ay,  5%  6%  88%  R   68% 

•  3%  Z-  7%  3l  3%  *  51%,  6-  97% 

*  18%  A  33%         8"  9%  9"  68%         la  83'/, 

The  reduction  of  the  following  complicated  expressions  will  afford  exercises  in 
addition,  subtraction,  multiplication,  and  division  of  fractions. 

1-  %  +  (2  %  of  2  %)  +  (%  X  2  %)  +  (6  %»+<%,) 

2.  4«/.x6%+-^  3.  ^kr  4-   "/8,X 


7  2%-l%  l8%5of73j 

%+%+%  „   fi3/  .  3-!%,  _        1 


1/       ,  1/       ,  1/  '«     %-%  6+1% 

/2%+/3%+/4% 

«•  1%,X6%  of22/wv_l%  .   oV^l+15% 

3 %-!%+< «  of  3  /•>     *     12  9"  3  /3  •  %+% 

,„   %+%+"/«.%  of  2%-hl "/„  ,>    15%-(4%Xl%) 

%+%+"/»       7»  of  4%-f-9%  "•     (V,+%,)+8»/» 


158.  Applications. — To  find  what  part  one  given  number  is  ot 
another. 

Example.— l.  What  part  of  8  is  3  ? 

Analysis. — 3  is  3/8  of  8  because  it  is  3  of  the  8  equal  parts  into  which  8  can 
be  divided.     Illustrate  by  the  use  of  counters. 

What  part  of 

2.  7  is  5  ?      16  is  12  ?     18  is  15  ?    21  is  14  ?    30  is  20  ? 

3.  39  is  26  ?    42  is  28  ?     48  is  36  ?     72  is  48  ?     32  is  24  ? 

Example.— 4.  What  part  of  5  is  %  ? 

Analysis. — 5  =  15/3,  and  2/3  is  2/15  of  15/:J  because  it  is  2  of  the  15  equal 
parts  into  which  15  thirds  can  be  divided. 

Note. — Illustrate  with  objects.     By  cutting  5  wholes  into  thirds,  and  taking  2 
of  the  fifteen,  we  see  how  nearly  this  problem  is  like  the  preceding. 


FRACTIONS. 


163 


What  part  of 

5.  4  is  %  ?  12  is  %  ?    15  is  %  ?    18  is  %  ?    25  is  %  ? 

6.  39  is  %  ?  16  is  %  ?    21  is  3/7  ?    28  is  %  ?    36  is  12/13  ? 
Example.— 7.  What  part  of  %  is  %  ? 

Analysis.- V5  =  12/i 5  and  */9  =  "/lBt  "/i,  is  »•/,',  of  12/is  because  it 
is  10  of  the  12  equal  parts  into  which  12  fifteenths  can  be  divided.  Illustrate 
with  objects. 

What  part  of 

.  8.  %is3/8?  %is%?  %is%?  %isy7?  5/nisy5? 

9.  3%  is  1%?  2%  is  1%?  4%  is  2%?  4%  is  »!/<| 
52/3is4y3? 

10.  6%  is   5%?     5%  is  43/7?     11   is  5%?     59/10  is  2%? 

r/2is6y3? 

159.  In  problems  such  as  the  preceding,  the  results  may  be 
reached  by  taking  the  number  representing  the  part  for  the 
numerator  and  the  number  representing  the  whole  for  the  de- 
nominator of  a  fraction,  and  reducing  as  suggested  in  Art.  157, 
or,  by  dividing  the  former  by  the  latter.     Thus  : 

l2/3  X  3  _  5 

2.  What  part  is  5%  of  7%?    Solution: 


l.  What  part  of  4  is  1%  ?        Solution: 


4X3 
53/4Xl2 

12 
69 

7%X12 

94 

SLATE     EXERCISES. 

What  part  of 

3.  42  is  3/7  ?     14  is  4/7  ?    30  is  %  ?    38  is  %  ?    45  is  %  ? 

4.  48  is  %f    33  is  %  ?    52  is  %  ?    48  is  %  ?    55  is  n/12  ? 

5.  y9isy3?  5is2y5?  4y12is3y6?  i%isiy7?  5y2is4y3? 

6.  4%  is  22/10?    7%   is  4%?    8%  is   5%?    8%  is   7%  ? 
1010/15  is  5%? 

7. 12  is  7%?  i2y3  is  ey3?    n%  is  y5?    y5  is  y35?  y. 


is  % 


8.  Vs  is  %  ?     V*  is  V„  ?     Vit  is  Vm  ?     1  Vn  is  % 


164 


STANDARD  ARITHMETIC. 


160.  From  a  known  fractional  part  of  a  number  to  find  the 
number. 

Example. — l.  6  is  %  of  what  number  ? 

Analysis. — If  6  is  3/4  of  a  number,  one  fourth  of  the  number  is  x/3  of  6  =  2, 
and  four  fourths  is  4  times  2  =  8. 

Example. — 2.  18/19  is  6/7  of  what  number  ? 

Analysis. — If  18/19  is  6/7  of  any  number,  1/7  of  the  number  is  a/6  of  18/19 


=  3/19  of  the  number,  and  7/7  or  the  entire  number  is  7  x  3/j 


7i9  =  i7i 


3.  %  is  %  of  what  number  ?  6.  %  is  11/1S  of  what  number? 

4.  7  is  3/8  of  what  number  ?  7.  5y8  is  y3  of  what  number  ? 

5.  4y2  is  9/10  of  what  number  ?        8.  62/3  is  5/6  of  what  number  ? 


Aliquot  Parts. 


161.   An  aliquot  part  of  a  number  is  any  integer  or  mixed 
number  that  will  exactly  divide  it  without  a  remainder. 

Aliquot  Parts  of  a  Dollar. 


50*   =*y8 

12%*=$% 

87%*=*% 

33%*=$y3 

m  =iyio 

75*    =•% 

25*  =$y4 

$  /st^tftB 

62%*=$% 

20*   =$y5 

6%*=l%8 

60*     =*% 

16%*=$% 

5*    =$y80 

37%*=$% 

162.  To  find  the  cost  of  a  number  of  articles  when  the  price 
is  an  aliquot  part  of  a  dollar. 

Example. — l.  What  will  20  doz.  eggs  cost  @  25*  a  doz.  ? 

Analysis.— At  $1  a  doz.,  20  doz.  would  cost  $20,  but  at  250  (=  J/4  dol.)  a 
doz.,  20  doz.  will  cost  l/4  of  $20,  or  $5. 

2.  What  is  the  cost  of  28  readers  @  25*  ?  %  50*  ? 

3.  144  lb.  of  beef  @  12l/8*  ?   @  16%*  ?   @  8%*  ? 

4.  240  lb.  of  raisins  @  20*  ?   @  16%*  ?   @  12%*? 

5.  48  pr.  of  socks  @  50*  ?   @  33%*  ?   @  37 y8*  ? 

6.  160  lb.  of  sugar  @  5*  ?   @  6%*  ?   @  8%*  ?   @  12%*  ? 


FRA  CTIONS. 


165 


7.  24  pr.  of  gloves  @  87*/**  ?   @  W  ?   @  62%$*  ? 

8.  35  handkerchiefs  @  600  ?  @  40^  ? 

9.  16  baskets  @  87>#  ?  @  62  %£  ?  @  87%*  ? 

10.  12  pr.  of  slippers  @  $1.25  ?  @  $1.33 %  ?  @  $1.16%  ? 

11.  230  bu.  of  wheat  @  62  7,*  ?   @  75^  ?   @  87  %^  ? 

12.  84  chairs  @  $1.25  ?   @  $1.33%  ?   @  $1.37%  ? 


163.  To  find  the  number  of  articles  that  can  be  bought  for  a 
given  sum,  when  the  price  of  one  is  an  aliquot  part  of  a  dollar. 

13.  At  33  y30,  how  many  lb.  of  butter  can  be  bought  for  $7  ? 

Analysis.— If  ttlf9f  will  bv:y  1  lb.,  $1  will  buy  3  lb.,  and  $7  will  buy  7  x 
3  lb.  =  21  lb. 

Thus  we  have  the  convenient 

Rule.— Multiply  the  number  of  articles  that  can  be  bought  for 
$1  by  the  number  of  dollars. 

14.  At  16%*,  how  many  books  can  be  bought  for  $9?    At 
33%?*? 

15.  At  25^,  how  many  pr.  of  cuffs  can  be  bought  for  $7.50  ? 

16.  At  33%#,  how  many  handkerchiefs  can  be  bought  for  $6  ? 
@  50^  ?    @  250  ? 

17.  For  $3.25,  how  many  qt.  of  milk  can  be  bought  @  6%0  ? 
@50?   @8%0? 


18.  Find  the  cost  of 

19.  Find  the  cost  of 

7  thermometers 

@      .50 

13  printing-presses 

<g>    $2.75 

9  thermometers 

®      .75 

5  velocipedes 

@    $7.75 

5  pr.  opera-glasses 

<&  $9.12V2 

7  canoes 

@  $55.75 

3  pr.  roller-skates 

@  $1.75 

31/4  bushels  potatoes 

@         .90 

72  scroll-saw  blades 

@       .20 

16  cwt.  rice 

@    $6.06V4 

13  pr.  pincers 

®    .  .35 

32  bushels  wheat 

@    $1.16g/3 

3  magnets 

©      .75 

12  sacks  salt 

@    $1.33V3 

18  pocket  compasses 

®  $1.162/3 

9  lb.  candy 

@        .37Va 

16  yards  silk  ribbon 

@      .871/. 

3  qt.  ice  cream 

@.        .400 

31/2  lb.  copper- wire 

@      .62V. 

2  bicycles 

@  $97.50 

1/8  lb.  fine  copper-wire  @.    $1.87V» 

64  lb.  wool 

@        .87Vf 

166  STANDARD  ARITHMETIC. 

Miscellaneous  Examples. 

ORAL     EXERCISES. 

1.  Multiply  the  numerators  of  %  %  %  3/8,  3/4,  %  %  7/8, 
by  3  ;  by  4 ;  by  5  ;  by  6.  How  do  these  multiplications  affect 
the  value  of  the  fractions  ? 

Note. — Frequent  illustrations  should  be  given  with  diagrams  and  counters. 

2.  Divide  the  numerators  of  %  %  %,  %,  %,  %,  %*  8/18, 
12/32,  by  2.  How  are  the  values  of  the  fractions  affected  by  these 
divisions  ? 

3.  Multiply  the  denominators  of  %,  %,  5/6,  %,  5/8,  7/8,  by  4 ; 
by  5  ;  by  10.  Are  the  fractions  increased  or  diminished  by  these 
multiplications  ? 

4.  Divide  the  denominators  of  4/4,  %,  4/12,  8/12,  8/m>  %4>  %2> 
by  2  ;  by  4.  How  are  the  values  of  the  fractions  affected  by 
dividing  their  denominators  by  integers  ? 

5.  State  four  ways  in  which  the  value  of  a  fraction  may  be 
changed,  and  give  examples  to  illustrate  each. 

6.  What  fraction  is  %  of  6%7  ?  Vio>   Vis?  %•  °^  ^ne  same  ? 

7.  By  how  much  is  %  greater  than  1/12  ?  Find  the  difference 
with  the  aid  of  slips  of  paper  or  parts  of  other  objects.  Tell  what 
you  do,  and  state  the  result. 

8.  Add  together  %  %  %  %  and  %  of  60. 

9.  Which  is  the  greater,  %  of  40,  or  %  of  30  ?    %  of  72  or 

9/n  of  77  ?     (Illustrate  with  counters.) 

10.  How  many  times  greater  is  y6  than  yl5  ?  y6  than  %^  ? 
%  than  %  ?  %  than  %  ? 

11.  Add  3  to  each  of  the  terms  of  3/7,  and  tell  how  much  the 
value  expressed  is  increased  or  diminished. 

12.  Invert  3/7  thus,  7/3 ;  add  3  to  each  term.  By  how  much 
is  the  value  p,f  this  fraction  increased  or  diminished  ? 

13.  From  75  subtract  7/i5  pf  75,  and  find  2/5  of  the  remainder. 

14.  What  is  the  difference  between  3/5  of  %  and  %  of  2/3  ? 


FRACTIONS.  1G7 

15.  Sixteen  is  %  of  what  number  ?  14  is  %  of  what  number  ? 

16.  A  train  starts  from  Indianapolis  at  7%  a.m.;  it  reaches 
Cincinnati  6%  hours  later.     At  what  o'clock  does  it  arrive  ? 

17.  Add  together  %  inches,  %  foot,  and  %  yard. 

18.  I  bought  16  oranges  for  24^.  .How  much  was  that  per 
dozen  ? 

19.  Man  ordinarily  spends  y3  of  his  time  in  sleep.  How  many 
hours  at  that  rate  does  he  sleep  in  a  fortnight  (14  days)  ? 

20.  If  I  read  2/5  of  a  book  in  a  day,  how  long  at  the  same  rate 
shall  I  be  in  reading  the  whole  of  it  ? 

21.  Eight  times  %  of  18=?  9  times  %  of  16=?  7  times 
%  of  15=? 

22.  I  had  $% ;  spent  $yi0  for  ink,  $y,  for  writing-paper,  and 
$y2o  for  pens.     How  much  money  had  I  left  ? 

23.  Eead  the  following  fractions  in  the  order  of  their  value, 
beginning  with  the  smallest :  7I^  7/6,  y^,  7/8,  7/12. 

24.  Eighteen  and  two  fifths  yards  are  cut  from  a  piece  of  cloth 
measuring  273/5  yd.     What  fraction  of  the  piece  remains  ? 

25.  Eive  ninths  is  2/3  of  what  number  ?  7/12  is  3/7  of  what 
number  ? 

26.  How  many  copper  wires  y30  of  an  inch  in  diameter  must 
be  laid  side  by  side  to  cover  y8  inch  ?    2/3,  5/6,  9/10  inch  ? 

27.  In  an  orchard  y6  of  the  trees  are  apple-trees,  yi2  pear,  y9 
plum,  y3  peach,  and  22  are  cherry-trees.     How  many  in  all  ? 

28.  Five  little  girls  held  a  fair  for  the  benefit  of  the  Fresh-Air 
Fund.  After  paying  out  for  expenses  yl0  of  the  whole  sum  re- 
ceived, they  contributed  $36  to  the  Fund.  How  much  did  they 
receive  ? 

29.  Early  June  peas  are  18^  a  can  at  retail.  How  much  do 
I  save  per  can  by  buying  them  at  wholesale,  $1.90  per  doz.  ? 

30.  The  sum  of  two  numbers  is  27/8.  One  of  the  numbers  is 
1%     What  is  the  other  ? 


168  STANDARD  ARITHMETIC. 

31.  Johnny  weeded  1/7  of  his  garden  on  Monday  ;  %  on  Tues- 
day ;  y3  on  Wednesday,  and  in  the  remaining  days  of  the  week 
finished  the  task  in  equal  portions.  What  part  did  he  do  each  of 
those  days  ? 

32.  Edith  earns  a  cent  for  each  extra  %  hour  she  practices 
her  music.  Last  week  she  earned  $y4.  How  many  extra  half- 
hours  did  she  practice  ? 

33.  A  woman  weaves  6  yards  cloth  in  2  days,  of  12  hours  each. 
What  part  is  the  work  of  1  hour?  If  she  is  paid  $%  a  yd., 
what  are  her  day's  wages  ? 

34.  A  painter  bought  18%  quarts  of  turpentine  at  y6  of  a  dol- 
lar per  qt.  He  sold  it  at  y4  of  a  dollar  per  qt.  What  did  it  cost 
him,  and  what  was  his  profit  ? 

35.  Divide  the  sum  of  2y3  and  S%  by  their  difference. 

36.  Twelve  yards  of  goods  3/4  yd.  wide  will  make  me  a  dress. 
How  many  yards  will  I  need  of  silk  that  is  yg  yd.  wide  ? 

37.  On  the  4th  of  July  Mr.  Brown  divided  %  of  $4.00  among 
his  children.  To  the  eldest  he  gave  J/4  of  the  %,  and  to  each 
of  the  others  45^.     How  many  children  had  he  ? 

38.  John  and  Will  together  mow  the  lawn.  John  mows  y4  in 
1  hour,  and  Will  '/§.     How  long  does  it  take  both  to  mow  it  ? 

39.  If  a  man  can  do  5/9  of  a  piece  of  work  in  4  days,  in  what 

time  will  he  do  the  entire  job  ? 

Analysis. — If  he  does  5/9  in  4  days,  he  will  do  1/9  in  1/5  of  4  =  4/5  day, 
and  9/9,  or  the  whole,  in  9  x  4/«  =  36/6  =  1 1/5  days. 

40.  If  4/n  of  an  acre  of  ground  yields  40  bushels  of  tomatoes, 
how  many  bushels  per  acre  ? 

41.  If  %  of  a  bushel  of  Bermuda  potatoes  costs  90^,  what  is 
the  price  per  bushel  ? 

42.  If  4/7  of  a  yard  of  velvet  costs  $5,  what  does  1  yard  cost  ? 

43.  Mr.  Jackson  sold  7/13  interest  in  his  shop  for  $5,600. 
What  was  the  whole  business  valued  at  ? 


FRACTIONS.  169 

44.  If  a  horse  can  trot  %  of  a  mile  in  1  minute,  in  what  time, 
at  the  same  rate,  can  he  trot  1  mile  ?    %  of  a  mile  ? 

45.  If  I  had  %  and  1/i  more  tulips  in  my  garden  I  should 
have  57.     How  many  have  I  now  ? 

46.  If  a  man  earns  $y3  per  hour,  a  woman  $y5,  and  a  boy 
$yi2,  what  do  all  receive  for  1  hour's  work  ? 

47.  How  many  hours  must  the  boy  work  to  receive  an  hour's 
wages  of  a  man  ?   How  many  the  woman  ? 

48.  In  what  time,  working  together,  can  the  woman  and  the 
boy  earn  an  hour's  wages  of  the  man  ? 

49.  What  number  diminished  by  %  and  %  of  itself  leaves  a 
remainder  of  30  ? 

50.  Seven  tenths  of  a  certain  number  less  %  of  it  is  15.  What 
is  the  number  ? 

51.  My  age  is  %  of  my  brother's ;  his  age  is  %2  of  father's, 
who  is  72  years  old.     How  old  am  I  ? 

52.  A  plague  carried  off  %  of  a  flock  of  sheep  in  one  week, 
%  of  the  remainder  the  next,  and  28  were  left.  What  was  the 
original  number  of  sheep  ? 

53.  A  contractor  was  to  receive  $60,000  for  a  building,  but 
forfeited  y40  that  amount  because  it  was  not  finished  within  the 
specified  time.     How  much  did  he  lose  ? 

54.  If  2  leaps  of  a  dog  are  equal  to  3  leaps  of  a  hare,  how 
many  leaps  of  the  dog  are  equal  to  27  of  the  hare  ? 

55.  What  number  is  reduced  to  64,  when  %  of  it  are  taken 
away  ?  

SLATE     EXERCISES.  _ 

1.  My  study  measures  14%  feet  in  length,  and  12%  feet  in 
width.     How  many  square  feet  in  the  floor  ? 

2.  A  money-bag  contains  37  half-dollars,  49  quarter-dollars, 
37  twenty-cent  pieces,  39  dimes,  63  nickels.  How  much  money 
in  all  ? 


170  STANDARD  ARITHMETIC. 

3.  An  employer  pays  $95  %  to  his  workmen,  each  one  re- 
ceiving $13  %.     How  many  are  there  ? 

4.  If  56  laborers  earn  each  $y3  per  hour,  how  much  do  they 
all  earn  in  6  days  and  6  hours,  reckoning  8  hours  to  the  day  ? 

5.  Twenty-nine  and  four  fifths  yards  were  sold  from  a  piece 
of  cloth  measuring  42%  yd.  What  was  the  value  of  the  re- 
mainder at  $liy20  a  yd.? 

6.  A  merchant  buys  52  %  bushels  of  beans  at  %\}J%  a  bu.; 
35%  bushels  of  peas  at  $1%  a  bu.;  28%  bushels  of  cranberries 
at  $2y3  a  bu. — (1)  Find  the  cost  of  each  item.  (2)  Find  the 
total  cost,  and  the  whole  number  of  bushels. 

7.  He  made  a  profit  of  iy80  on  every  quart  of  beans,  2y3^  on 
every  qt.  of  peas,  and  2y6^  on  every  qt.  of  cranberries. — (1)  Find 
the  profit  on  each  item.     (2)  Find  the  total  profit. 

8.  A  farm  of  276%  acres  rents  for  $2013 %  What  is  the 
rent  per  acre  ? 

9.  I  bought  45  government  bonds  at  $105 1/2,  and  sold  them 
at  $106  7/8.     Find  the  gain. 

10.  A  house-painter  earns  $3%  a  day  of  10  working  hours. 
One  week  in  which  he  worked  extra  time  his  pay  amounted  to 
$25.     How  many  extra  hours  did  he  work  that  week  ? 

11.  What  will  7%  yards  of  lace  cost  at  $%  per  yd.  ?    At  $%  ? 

At  $%?  At  $y12? 

12.  A  person  standing  exactly  under  the  equator  is  carried  by 
the  rotation  of  the  earth  24,899  miles  a  day.  How  many  miles 
is  he  carried  in  1  hour,  2  h.,  3  h.,  5  h.,  6  h.,  8  h.,  12  h.  ?  (What 
part  of  a  day  is  1  hour  ?     Do  the  work  with  as  few  figures  as  possible.) 

13.  If  7%  lb.  coffee  cost  $2yi0,  what  will  11%  lb.  cost? 
10%  lb.?    4%  lb.?    12%  lb.? 

14.  Mr.  A.  left  by  will  %  of  his  estate  to  his  wife,  %  of  the 
remainder  to  his  eldest  son,  y3  of  what  was  then  left  to  his  eldest 
daughter,  and  $20,000  to  each  of  his  two  other  children.  What 
was  the  value  of  the  estate  ? 


FRACTIONS.  171 

15.  A  grocer  bought  two  tubs  of  butter,  weighing  together 
70u/12  lb.  One  tub  when  empty  weighed  7y4  lb.,  and  the  other 
8y3  lb.     How  much  butter  did  he  buy  ? 

16.  Find    the   sum    and    the    difference   of    (33/4-i-52/5)   and 

17.  Find  the  sum  and  the  difference  of  (54/5x36/7)  and 
(7%X3%). 

18.  If  it  takes  a  workman  %  of  a  day  to  do  %  of  a  piece  of 
work,  how  much  of  it  can  he  do  in  5/8  of  a  day  ?  How  much  in 
3/7  of  a  day  ? 

19.  If  %  of  an  acre  of  land  is  sold  for  $45  3/20,  what  is  the  re- 
mainder worth  at  double  the  rate  ? 

20.  If  yi5  of  a  box  of  merchandise  is  worth  $7%,  what  is  % 
worth  ? 

21.  A  grocer  mixes  57 %  lb.  of  tea,  at  $6/10  a  lb.,  with  42 1/2  lb. 
of  tea  at  $7/10  a  lb.     What  is  the  value  of  a  lb.  of  the  mixture  ? 

22.  A  farmer  sold  5/8  of  his  wheat  at  ll1/™  a  bushel,  and  re- 
ceived $796  %  for  it.  How  many  bushels  did  he  sell,  and  how 
many  did  he  have  at  first  ? 

23.  Mr.  Hill,  having  $500  to  pay  expenses,  made  a  journey 
that  lasted  6  weeks.  On  reaching  home  he  had  $46%  left. — 
(1)  How  much  did  he  spend  ?  (2)  What  was  the  average  ex- 
penditure per  week  ? 

24.  I  bought  a  house  and  paid  down  y3  of  the  price,  and  in 
one  year  thereafter  I  paid  %  of  the  price.  The  two  payments 
amounted  to  $43,780.     What  was  the  price  of  the  house  ? 

25.  A  clerk  has  a  monthly  income  of  $75,  and  spends  $542/5 
per  month.     How  much  does  he  save  a  year  ? 

26.  By  how  much  would  he  have  to  diminish  his  expenses,  per 
month,  to  save  $20 1/2  per  year  more  than  he  now  does  ? 

27.  A  laborer  borrowed  from  his  employer  $66  %0,  agreeing  to 
pay  it  by  having  $245/100  deducted  from  his  wages  every  week. 
How  many  weeks  at  that  rate  did  it  take  him  to  pay  his  debt  ? 


172  STANDARD  ARITHMETIC. 

28.  If  %  of  7  lb.  of  coffee  costs  $7/8,  how  many  lb.  can  be 
bought  for  $l23/25  ?    $2%  ?    $5%  ? 

29.  What  is  the  sum  of  the  area  of  5  fields,  containing  severally 
93 %,  24%,  86 »/M,  56%,  and  89  %4  acres? 

30.  What  is  the  cost  of  23%  lb.  flour  at  $%•  ?  15%  lb.  oat- 
meal at  $%5?  3%  lb.  raisins  at  $%5  ?  17%  lb.  nails  at  4^? 
1  doz.  fire-shoyels  at  12  %£  apiece  ? 

31.  What  number  multiplied  by  37/8  will  give  2 ;  what  num- 
ber divided  by  it  will  give  %  ? 

32.  What  number  multiplied  by  %  of  ll3/4  will  produce  1. 

33.  In  a  school  of  100  pupils,  of  whom  3/5  are  boys,  7  boys  and 

4  girls  are  absent.     What  part  of  the  boys  are  present  ?    What 
part  of  the  girls  ? 

34.  One  third  of  the  eighth  part  of  what  number  is  equal 
to  9%? 

35.  How  many  cubic  feet  in  a  box  4%  ft.  long,  3%  ft.  wide, 

7%  ft.  deep.  ?     (See  problems,  page  103.) 

36.  If  one  faucet  empties  a  cistern  in  6  hours,  and  another  in 
9  h.,  in  what  time  will  both  together  empty  it  ?  What  part  of 
the  contents  will  the  two  faucets  discharge  in  1  hour  ? 

37.  In  what  time  will  both  empty  it  if  the  first  begins  to  run 
after  the  second  has  run  for  2  hours  ? 

38.  A  can  set  the  type  for  a  certain  book  in  6  days,  B  in  8, 
C  in  9,  and  D  in  12  days.  In  what  time  can  they  do  it  working 
together  ?     (What  part  will  they  all  do  in  a  day  ?) 

^    39.  How  long  must  a  room  4%  yards  wide  be  to  contain  as 
many  square  yards  in  the  ceiling  as  a  room  7%  yards  long  and 

5  yg  yards  wide  ? 

40.  I  can  walk  20  miles  in  5  hours,  and  my  friend  can  do  it 
in  6  hours.  Starting  at  the  same  time  from  points  20  miles 
apart  and  walking  toward  each  other,  how  far  are  we  apart  in  1 
hour,  and  in  what  time  from  starting  would  we  meet  ? 


CHAPTER  X. 

DECIMAL   FRACTIONS. 


164.  The  last  chapter  presented  a  mode  of  writing  fractions 
in  which  the  number  of  parts  are  indicated  by  one  number  and 
their  names  by  another.  This  chapter  shows  how  both  the  num- 
ber and  name  of  certain  fractional  parts  may  be  represented  by 
the  decimal  system. 

Note. — Exercises  on  the  following  diagram  are  designed  to  familiarize  the  pupil 
with  the  relations  of  such  parts.  Bundles  of  jackstraws  will  also  serve  for  illus- 
tration. 


174 


STANDARD  ARITHMETIC. 


Illustration. — If  a  square  sheet  of  paper  were  ruled 
into  10  long  slips,  and  each  of  these  were  subdivided  into 
10  small  squares,  and  the  small  squares  into  10  short  slips, 
and  each  short  slip  into  10  tiny  squares,  as  shown  in  the 
two  slips  below : 

1.  How  many  long  slips  would  there  be  ? 
How  many  small  squares?  How  many  small 
slips  ?    How  many  tiny  squares  f 

2.  What  part  of  the  whole  diagram  is  a 
long  slip?  A  small  square?  A  small  slip? 
A  tiny  square  ? 

3.  What  part  of  a  long  slip  is  a  small  square  ? 
A  short  slip  ?  A  tiny  square  ?  What  part  of  a 
small  square  is  a  tiny  square  ?  etc.?  etc. 


Note. — The  questions  given  above  are  only  suggestive  of  exercises  designed  to 
make  the  pupil  familiar  with  decimal  parts  and  their  relations. 

165.  The  division  of  anything  into  ten  equal  parts,  and  the 
subdivision  of  these  into  ten  smaller  equal  parts,  and  so  on,  are 
Decimal  Divisions,  and  the  parts  are  Decimal  Parts. 

Note. — The  dime  is  a  decimal  part  of  a  dollar,  the  cent  a  decimal  part  of  a 
dime,  the  mill  a  decimal  part  of  a  cent. 

166.  A  Decimal  Fraction  is  one  or  more  of  the  decimal 
parts  of  a  unit. 


DECIMAL  FRACTIONS.  175 

Decimals  expressed  in  Figures. 

167.  The  first  illustration  (page  173)  represents  421  sheets 
of  paper,  and  2  tenths,  3  hundredths,  4  thousandths,  5  ten  thou- 
sandths of  a  sheet ;  and  as  each  figure  of  421  indicates  by  its 
place  whether  it  represents  units,  tens,  or  hundreds,  so  the  figures 
2,  3,  4,  and  5  may  be  made  to  indicate  by  their  places  whether  they 
represent  tenths,  hundredths,  thousandths,  or  ten-thousandths. 
But  to  show  that  they  represent  parts  and  not  wholes,  that  they 
are  decimals  not  integers,  a  point,  called  the  decimal  point  (.), 
is  placed  before  them,  and  the  number  is  written  thus  :  421.2345. 

168.  Tlie  cipher  is  used  in  decimal,  as  in  integers,  to  mark 
vacant  places.  Thus,  if  the  two  long  slips  were  omitted  in 
the  illustration,  the  number  represented  would  be  expressed  by 
421.0345.  If  there  were  no  long  slips  nor  small  squares  it  would 
be  written  421.0045,  etc.,  etc. 


EXERCISES    ON     DIAGRAM. 

1.  How  many  sheets  and  how  many  and  what  parts  of  a  sheet 
are  represented  by  4.2?  2.05?  3.82?  .35?  .23?  1.01?  2.71? 
.182?  .19?  41.41?  3.00?  .4321?  10.1?  7.15?  6.01?  .101? 
17.208?  15.001?   21.0021?  .0053? 

Give  first  the  descriptive  names  of  the  parts,  as  long  slips,  small  squares, 
etc.,  then  use  the  proper  arithmetical  terms,  tenths,  hundredths,  etc.,  thus:  4  sheets 
and  2  long  slips,  or,  4  sheets  and  2  tenths  of  a  sheet. 

2.  Illustrate  by  diagram,  on  slate  or  blackboard,  what  is  meant 
by  .01,  by  .25,  by  .35,  by  3.7,  by  1.3,  by  2.004,  etc. 

3.  Is  there  any  difference  in  value  between  6.7  and  6.70  ?  Be- 
tween 3.7  and  3.07  ?    Between  5.16  and  50.16  ?    Between  .81  and 

.8100  ?      (In  stating  the  differences,  tell  what  parts  of  the  diagram  are  repre- 
sented in  each  case.) 

4.  Tell  how  many  long  slips,  small  squares,  etc.,  must  be  cut 
from  a  sheet  of  paper  to  have  .357  of  a  sheet  ?  To  have  .5642  ? 
To  have  .045?  etc. 


176  STANDARD  ARITHMETIC. 

Without  the  aid  of  words,  express  in  figures  the  number  of 
sheets  and  parts  of  sheets  described  below,  and  read,  using  the 
proper  decimal  terms : 

1.  207  sheets  7  long  slips  and  8  tiny  squares ;  3  small  squares 

5  short  slips  and  5  tiny  squares. 

2.  10  sheets  and  1  tiny  square ;  75  sheets  and  1  short  slip  ; 

6  sheets  and  6  small  squares. 

3.  17  sheets  7  hundredths  and  9  thousandths  of  a  sheet ;  13 
sheets  and  3  ten  thousandths. 

4.  24  sheets  3  tenths  and  6  thousandths ;  8  thousandths  and 

7  ten  thousandths. 

Note. — We  can  represent  1/3,  e/6,  3/7,  or  other  common  fraction  of  a  decimal 
part,  by  writing  the  common  fraction  after  the  decimal,  thus  :  .2  1/3  is  read  2  x/3 
tenths.  5  3/7  small  squares  would  be  expressed  by  .05  3/7,  which  is  read  5  3/7 
hundredths.     .63/7  is  6  tenths  and  3/7  of  a  tenth. 

Without  the  aid  of  words,  express  in  figures  : 

1.  3  sheets  8  %  short  slips  ;  7  sheets  6  y3  small  squares. 

2.  13  sheets  8  2/7  slips ;  23468  sheets  5  %  small  squares. 

3.  25  sheets  4%  hundredths  ;  81  sheets  9%  thousandths. 

4.  86  sheets  5%  tenths ;  4000  and  73/7  ten  thousandths. 

5.  Which  has  the  greatest  value,  the  1/2  in  21/2,  .2y2,  or  .02y>  ? 

Note. — We  may  represent  entire  decimal  parts  by  fractions  written  in  the  com- 
mon fractional  form,  thus:  3  small  squares  may  be  represented  by  3/i0o- 

6.  Tell  how  many  whole  sheets,  long  slips,  etc.,  are  repre- 
sented by  the  following  figures  :  15/10,  2yi00,  %,  38/100o,  VlOO, 
Viooo,  20%*  120yi0,  999/100,  37y100,  43/10. 

7.  Write  the  following  fractions  in  decimal  form  :  17/10  (=  1.7), 

131/  3468/  2426/  1769/  4432/         1286/  316/  K  7/        98  29/ 

/ioo>         /ioo>        /ioooj         /loot)?         /io>        /loooj       /ioo>    °  /m  ^°    /ioooo> 

OJ.36/  K46/  71243/  -\(\£5f         111/  19/ 

°*    /100,    O     /i000>  /lOOOOj     1KJ^=  flOt  /iooo>        /10« 

Note. — Any  fraction  having  for  a  denominator  10,  100,  1000,  etc.,  is  properly 
a  decimal  fraction,  because  it  represents  parts  obtained  by  the  division  of  the  unit 
into  tenths,  tenths  of  tenths,  etc.,  etc.  But  the  term  decimal  is  used  alone  only 
when  there  is  no  denominator  expressed. 


DECIMAL  FBACTIONS.  .      177 

Definitions. 

169.  A  Decimal  point,  or  sign  (.),  is  a  period  prefixed  to  a 
decimal  to  distinguish  it  from  an  integer. 

170.  A  Pure  Decimal  consists  of  decimals  only. 

171.  A  Mixed  Decimal  is  one  that  consists  of  an  integer  and 
a  decimal. 

172.  A  Complex  Decimal  is   one   consisting   of  a  decimal 
with  a  common  fraction  annexed. 


Decimal  Table. 

173.  The  following  table  will  facilitate  the  learning  of  the 
several  orders.  The  correspondence  between  the  names  of  the 
places  to  the  right  and  left  of  units  should  be  noticed. 

Table. 

S5  tf 


I.  Ill  ,       I  ill  3ii 


9  ff 


Names.        ^  §     'SIS     J  .     .?      gj'f      §^ 

£  J»  K     i^sfl     lS  J5    f>    §^     f  r§  v2     22  J  ^ 


537      290      32     7.03      214      516 


Integers.  Decimals. 


Reading  Decimals  in  Terms  of  the  Lowest  Order. 

1.  2  long  slips  and  3  small  squares,  make  how  many  small 
squares  ? 

2.  2  tenths  and  3  hundredths,  make  how  many  hundredths  ? 

3.  5  long  and  4  short  slips,  make  how  many  short  slips  ? 


178  STANDARD  ARITHMETIC. 

4.  2  tenths,  3  hundredths,  and  4  thousandths,  make  how  many 
thousandths  ? 

5.  2  long  slips,  3   small  squares,  4  short  slips,  and   5   tiny 
squares  =  how  many  tiny  squares  ? 

6.  2  tenths,  3  hundredths,  4  thousandths,  and    5  ten-thou- 
sandths =  how  many  ten-thousandths  ? 

Hence  for  reading  decimals  we  have  the 

1 74-.   Rule. — Read  the  decimal  as  if  it  were  a  whole  number, 
and  give  it  the  name  of  the  right  hand  order. 

Thus,  .3567  is  read  3567  ten-thousandths;  .169  as  169  thousandths;  .354789 
as  354789  millionths. 

ORAL    EXERCISES. 

1.  Eead,  .1;  .6;  .9;  .45;  11.4;  13.47;  51.67;  6.15;  8.24; 
98.34;  100.1;  345693.71. 

2.  Eead,  100.73;  27.02;  50.57;  6.67;  41.01;  120.03;  200.01. 

3.  Eead,  1.111;  .567;  .004;  75.123;  3.004;  1.012;  6.953. 

4.  Eead,  92.009;  9.00012;  13.8947;  57.625341;  1.06777893. 


5.  Eead,  pronouncing  separately  the  order  of  each  digit  in  the 

fractional  parts  :   61.43  (61  and  4  tenths,  3  hundredths). 

10.9;  738.5423;  4.02;  5.063;  31.02803;  39.417356;  10.1324; 
12.11;    26.103;    17.1101;    29.922;     30.87203456;    9.39485762. 


6.  Eead  the  following  as  mixed  decimals,  that  is,  the  units  first 

then  the  decimal  :  (Read  6.12  thus,  six  and  12  hundredths). 

14.013;  6.57;  3.0154;  46;  1044;  9.999;  20.02;  35.04; 
46.34256;       50.148735;       83.4283;       87.87328;      7.5983. 

7.  Eead  the  following  as  improper  fractions,  that  is,  read 
integer  and  fraction  together  as  one  number,  giving  to  the  whole 
the  name  of  the  lowest  decimal  order.     (Read  7.04  as  704  hundredths.) 

18.164;  516.2;  5.005;  29.092;  5.79;  13.579;  1357.9;  1.010; 
263.4501;  63.4;  63.04;  63.004;  63.0004;  5.00013.  (The  last  is 
read,  five  hundred  thousand  thirteen  hundred  thousandths.) 


DECIMAL  FRACTIONS.  179 

Suggestion. — Ask  yourself  whether  it  is  true  that  '7.04  is  equal  to  704  hun- 
dredths. Turn  to  the  illustration,  on  page  173,  and  study  this  out  for  yourself. 
How  many  small  squares  in  7  sheets  of  card-board  ?  How  many  in  7  sheets  and 
4  small  squares  ?  

8.  If  .  04  were  written  in  the  form  of  a  common  fraction,  what 
would  the  numerator  be  ?  What  the  denominator  ?  Answer  like 
questions  with  regard  to  the  decimals  in  exercises  1  and  2. 

Note  1. — Observe  that  when  the  denominator  is  written,  the  decimal  point  and 
the  ciphers  preceding  the  first  significant  figure  are  omitted  in  the  numerator :  thus, 

3/ 

.03  =  3/ioo>  not  ,03/ioo-     (,03/ioo  is  equivalent  to  the  complex  fraction  -r^Sf-) 

Note  2. — Observe,  also,  that  the  denominator  of  a  decimal,  when  written,  con- 
tains as  many  0's  as  there  are  figures  in  the  decimal. 


Writing  Decimals. 

For  the  writing  of  decimals,  the  following  rule  will  be  found  serviceable.  Skill 
is  to  be  obtained  only  by  practice. 

175.  Rifle. — Place  the  decimal  point,  then,  after  considering 
how  many  places  are  needed  to  give  the  last  figure  of  the  deci- 
mal its  proper  order;  write  each  figure  in  the  order  to  which  it 
belongs. 

Example.— Write  375  hundred  thousandths. 

Remembering  that  hundred  thousandths  is  the  fifth  decimal  order,  and  observ- 
ing that  375  contains  only  three  figures,  we  perceive  that  two  orders  must  be  filled 
with  ciphers,  thus :  .00375. 

SLATE    EXERCISES. 

Write  in  figures : 

1.  Three  and  fifteen  hundredths  ;  thirty-one  thousandths. 

2.  One  and  one  thousandth  ;  twelve  and  fifteen  hundredths. 

3.  One  hundred  twenty  eight  and  seventeen  thousandths. 

4.  Seventy-eight  ten  thousandths  ;  seven  hundredths. 

5.  Sixty-one  hundred  thousandths ;  ten  and  one  ten  thou- 
sandth. 

6.  Fifty-four  thousand  and  fifty-four  ten  thousandths. 

7.  Five  thousand  seventy-five  millionths. 


180  STANDARD  ARITHMETIC. 

Addition  of  Decimals. 

176.  Rule.— Write  the  numbers  to  be  added  so  that  figures 
of  the  same  order  shall  stand  in  the  same  column. 

Add  as  in  integers,  and  place  the  decimal  point  in  the  sum 
directly  under  the  decimal  points  in  the  numbers  added. 

Examples. 


1.  3.523 

2.     .9374 

3.     .12 

4.   5.678934 

23.42 

13.21 

5.2 

2.16674 

6.006 

45.135 

134.56 

.00374 

4.734 

1.0006 

42.03 

17.00003 

5.  6.6+77.77+888.888+26. 742+1.2+5.401+.002= 

6.  4.1535+.92+12.3472+.006+11.3+2. 00046+9.07= 

7.  100.2+59.012+8. +3.1205+69. +63.109+934563.4= 

8.  604. 1+.  012+18. 069+9. 232+8. 01+2. 10004+3. 05 = 

9.  10.901+12. +43.321986+.79342+4283.4132+6.7= 

10.  11.  12.  13. 

14.  14.3     +2.348  +  4.56   +17.01     +384.9000     = 

15.  9.58  +8.71   +6.54  +     .004  +   15.401       = 

16.  73.374  +  9.234  +  3.042+   9.345  +     3.789346= 

17.  1.583  +  5.006  +  7.1     +  7.2003  +  100.007384= 

Test  the  accuracy  of  your  results.     (See  note,  page  32.) 


Applications. — l.  Add  the  following  sums  of  money  :  $28.36, 
$108.09,  $27.50,  $1.30,  $38,742,  $387,655,  $998,999,  $3.27. 

2.  Six  marble  blocks  weigh  respectively  5.73  cwt.,  4.834  cwt., 
7.938  cwt.,  6.4  cwt.,  15  cwt.,  and  387.1  cwt.     Find  the  total 

weight. 

3.  A  train  on  the  Pennsylvania  R.  R.  ran  56.3  miles  in  the 
first  hour,  62.34  miles  in  the  second,  59.247  in  the  third,  60.7304 
in  the  fourth.     How  many  miles  altogether  ? 

4.  A  draper  bought  2  pieces  of  buckskin,  each  containing 
56.34  yards ;  2  pieces  of  rep,  each  containing  96.05  yards;  and  1 
piece  of  broadcloth,  containing  27.2  yards.  Find  the  number  of 
yards  in  the  5  pieces. 


DECIMAL  FRACTIONS.  181 

Subtraction  of  Decimals. 

177.  Rule, — Write  the  subtrahend  under  the  minuend,  so  that 
figures  of  the  same  order  shall  stand  in  the  same  column. 

Subtract,  as  in  integers,  and  place  the  decimal  point  in  the 
remainder  directly  under  the  decimal  points  of  the  minuend  and 
subtrahend. 


10. 


Examples.— l.  94.324 

2.  73.6           3.  5.4          4.  9.7            5.  6.01 

7.86    • 

19.79              4.38              6.543             3.4 

6.  7384.02             7.  9.004 

8.  3.28764           9.  15.60003004 

56.934                7.2043 

1.00009                    .794569376 

1.01          11.  4.003         12. 

15.                 13.  70.                 14.  50009. 

.09                2.006 

6.3785                16.7345                        5.0009 

15.       8.452-3.1052= 

21.  73845.009-1.23456  = 

16.    92.8245—9.86543= 

22.    9384.708-2.3457= 

17.        .0052— .0041  = 

23.       342.5703-.1994= 

18.      3.004— .0097= 

24.     6534.70045-3.7634= 

19.  121.12-8.943=       ' 

25.      897.309—3.1073= 

20.  423.4567382—413.05  = 

26.      328.00019—6.0004= 

Applications.  —  l.  From  a  lot  containing  10,000  □  yards, 
437.296  □  yds.  are  sold.     How  large  is  the  remaining  part  ?    (The 

sign  a  is  used  for  the  word  "  square.") 

2.  From  17.256  tons  of  coal  5.625  tons  were  nsed.  How  much 
was  left  ? 

3.  Mr.  Smith's  property  amounted  to  $47,300.75  when  he 
died.  Accounts,  to  the  amount  of  $340.95,  were  presented  and 
paid.     How  much  was  left  to  the  heirs  ? 

4.  The  French  meter  is  39.37079  inches.  How  much  longer 
than  a  yard  is  the  meter  ? 

5.  Find  the  difference  in  height  of  two  flag-staffs,  the  one 
measuring  38.75  ft.,  the  other  53.9  ft. 

6.  Find  the  difference  between  .57  and  .7;  between  eight 
hundred  fifty-two  ten-thousandths  and  1. 


182  STANDARD  ARITHMETIC. 

Multiplication  of  Decimals. 
Example.— l.  Multiply  .75  by  3.     (Find  3  times  .75.) 

Process.  Analysis. — 3  times  5  hundredths  =  15  hundredths  =  1  tenth 

^5  and  5  hundredths;  3  times  *l  tenths  =  21  tenths;  21  tenths  +  1 


3 


tenth  =  22  tenths  =  2  units  and  2  tenths. 

Repeat  the  analysis,  using  the  terms  small  squares,  long  slips, 
2.25  and  sheets,  respectively,  for  hundredths,  tenths,  and  units. 

The  Multiplier  a  Decimal. 

Example. — 2.  Multiply  .75  by  .3.      (Find  3  tenths  of  75  hundredths.) 

Analysis. — 3  tenths  of  5  hundredths  =15  thousandths  =  1 

Process.         hundredth  and  5  thousandths ;    3  tenths  of  7  tenths  =  21  hun- 

#75  dredths;    21    hundredths  +  1   hundredth  =  22    hundredths  =  2 

o  tenths  and  2  hundredths. 

— '—  Repeat  the  analysis,  using  the  descriptive  terms  short  slips,  etc. 

.225  Thus,  3/10  of  5  small  squares  =  15  short  slips  =  1  small  square 

and  5  short  slips,  etc. 

Example. — 3.   Multiply  .75  by  .03.      (Find  3  hundredths  of  .75.) 

Process.  Analysis.  —  3   hundredths   of   5   hundredths  =  15  ten  thou- 

-,-  sandths  (see  diagram,  page  174) ;  15  ten  thousandths  (tiny  squares) 

=  1  thousandth  and  5  ten  thousandths ;  3  hundredths  of  7  tenths 

•0<3  =  21  thousandths ;    21   thousandths  +  1    thousandth  =  22  thou- 

.  0225  sandths  =  2  hundredths  and  2  thousandths. 

Repeat  the  analysis,  using  the  descriptive  terms  tiny  squares,  etc. 

178.  Thus,  we  find  that  if  the  order  of  the  multiplier  is  units, 
the  order  of  the  product  is  the  same  as  that  of  the  multiplicand. 
If  the  multiplier  is  tenths,  the  order  of  tbe  product  is  one  degree 
lower  ;  if  it  is  hundredths,  the  order  of  the  product  is  two  degrees 
lower,  etc. 

179.  Hence,  in  the  product  of  two  decimals  there  are  as  many 
decimal  places  as  there  are  in  the  multiplicand,  plus  the  number 
of  decimal  places  in  the  multiplier. 

180.  Rule.  —  Multiply  as  in  whole  numbers,  and  from  the 
right  of  the  product  point  off  as  many  figures  for  decimals  as 
there  are  decimal  figures  in  the  multiplier  and  multiplicand  to- 
gether. If  there  be  not  so  many  figures  in  the  product,  supply 
the  deficiency  by  prefixing  ciphers. 


DECIMAL  FRACTIONS.  183 


ORAL    EXERCISES. 

4. 

5. 

6. 

7. 

8. 

.2x3= 

14x.6  = 

.42x3  = 

.2x.3  = 

.2x.003  = 

.4x6= 

13  x. 5  = 

.26x5= 

.3x.6= 

.9x.005  = 

.5x5= 

15  x. 4= 

.31x7= 

.4x.5= 

.5x.008= 

.7x2  = 

17x.3  = 

.02x9= 

.5x.2= 

.7x.004= 

.8x4= 

16  x. 2= 

.63x8: 

.6x.4= 

.3  x  .007= 

SLATE     EXERCISES. 
Note. — It  is  well  for  pupils  to  accustom  themselves  to  estimate  results;  for 
instance,  if  it  is  required  to  multiply  5.65  by  7.001,  they  should  be  able  to  say  at 
a  glance  that  the  product  will  be  about  39,  that  is,  a  little  more  than  5  */i  times  7. 


9. 

10.                            11. 

12. 

.3x.093= 

.5  x  .934=             .52  x  .213= 

1.5  x. 3= 

.8x.075  = 

.7x.825=            .73  x. 332= 

2.4  x. 5  = 

.5  x  .084= 

.9x.738=            .84  x. 252= 

1.5  x. 7= 

.9x.063= 

.6x.225=             .95  x. 163= 

3.2  x. 6= 

.7x.052= 

.3x.367=            .62  x. 421  = 

1.6  x. 9= 

13. 

736.045  x.  843 

18.  .0009  x. 0543 

23.  84.008  x  1000.4 

14. 

93  x  .0067 

19.  9.00134x8.004 

24.  258.01  x  3030.1 

15. 

4.709  x  .7635 

20.  .195  x. 00027 

25.  .98x36.0007 

16. 

84.008  x  100.001 

21.  .1825  x  18.24 

26.  .357x88345.4 

17. 

17827.032  x  8.754 

22.  .75  x  .30052 

27.  28.601  X  3.425 

28.  Multiply 

.004:   .71;   .70014;   1.04  b} 

'  .0091. 

29.          " 

.05 

;  .17;  .999;   .7534  by  . 

0008. 

30. 

1000;   100;   .001;   .64;  .01 

by  2.847. 

Applications. — l.  My  age  is  1.075  times  my  brother's  ;  if  he  is 
30,  how  old  am  I  ?     If  he  is  25,  how  old  am  1  ? 

2.  What  is  the  area  of  a  lot  which  is  9.34  yd.  wide  and  48.5 

yd.  deep  ?     (How  many  square  yards  in  it  ?) 

3.  Find  the  area  of  a  field  .876  miles  by  .0056  miles  ? 

4.  At  $5.87  per  acre,  what  is  the  rent  of  a  farm  of  47.9  acres  ? 

5.  If  I  buy  2  cwt.  66  lb.  sugar  at  $13.09  per  cwt.,  and  sell 
it  at  $.12  per  lb.,  what  do  I  gain  or  lose  on  the  whole  ? 


184  STANDARD  ARITHMETIC. 

Division  of  Decimals. 
Example. — l.  How  many  times  .18  in  54  ? 

process<  Analysis. — In  54  units  there  are  5400  hundredths,  and  18 

1         .      _  hundredths  are  contained  300  times  in  5400  hundredths. 

- — - — -  Illustration. — 18  of  the  small  paper  squares  represented  on 

oOU.  page  174  can  be  taken  300  times  from  54  sheets. 

2.  How  many  times  .18  in  5.4  ? 

p  Analysis. — In  5.4  there  are  54  tenths  =  540  hundredths. 

'  In  540  hundredths,  18  hundredths  is  contained  30  times. 
I— '- —  Illustration. — Show  that  18  small  squares  are  contained  30 

30.  times  in  5  sheets  4  long  slips. 

3.  How  many  times  .18  in  .54?    Ans.,  3. 

4.  How  many  times  .018  in  54  ?    In  5.4  ?    In  .54  ? 

1 8 1  ■  Let  it  be  observed  that  in  every  case  the  dividend  must  be  reduced  to 
an  order  at  least  as  low  as  that  of  the  divisor.  Evidently,  if  we  are  to  ascertain 
how  many  times  18  short  slips  there  are  in  any  number  of  sheets,  long  slips,  or 
small  squares,  we  must  first  ascertain  how  many  short  slips  there  are.  Hence,  in 
division  of  decimals,  there  must  always  be  as  many  decimal  places  in  the  dividend 
as  in  the  divisor. 

5.  How  many  times  1.08  in  .05778  ? 

Process,  Explanation. — Beginning  with  tenths,  we  count  off  as 

.  035  many  decimal  places  in  the  dividend  as  there  are  in  the  di- 

1  ao\  AKiiywo  visor,  and  separate  them  from  the  places  to  the  right  by  a 

'*      '  short  vertical  line.     This  marks  the  point  below  which  no  in- 

54:0  teger  can  be  obtained  in  the  quotient  (no  quantity  can  be 

378  contained  any  whole  number  of  times  in  a  quantity  less  than 

094.  itself).     Here  also  the  decimal  places  must  begin,  for,  though 

.  one  tenth  of  the  divisor  be  not  contained  in  the  next  partial 

540  dividend,  the  place  must  be  marked  by  a  cipher  in  order  that 

54  }  figures  of  lower  orders  may  have  their  proper  places. 

(82.  Rule — 1.  Annex  ciphers  to  the  dividend,  if  necessary, 
till  the  right  hand  order  is  the  same  as  that  of  the  right  hand 
figure  of  the  divisor. 

2.  Divide  as  in  simple  division.  Place  the  decimal  point  imme- 
diately before  the  quotient  figure  that  is  obtained  from  the  order 
of  the  dividend  next  lower  than  the  lowest  order  of  the  divisor. 

Note. — There  must  always  be  as  many  decimal  places  in  the  quotient  as  there 
are  in  the  dividend  more  than  in  the  divisor. 


DECIMAL  FRACTIONS. 


185 


ORAL     EXERCISES. 


1. 

2. 

3. 

4. 

.4-f-8= 

2-f-.5  = 

.64-4-.08= 

.164-. 8= 

.64-5  = 

5-4-.8= 

.494-.07= 

.14-4-.7= 

.8-4-4= 

4-4-.6  = 

.364-.  03= 

.084-.4= 

.24-2= 

5-s-.7= 

.844-.04= 

,09-h.  3  = 

SLAT  E     EXERCISES. 

1.  7.32-T-6 

6. 

.964-32 

11.  14-.0037 

16.  10004-.09 

2.  123-^-6 

7. 

.16-k4 

12.  104-.001 

17.  .0045-4-9 

3.  127-f-6 

8. 

.58-^-3.1 

13.  .5^-1000 

18.  .03-S-1.004 

4.  4-^-. 008 

9. 

.63684-8 

14.  45.984-10 

19.  .03754-.03 

5.  4.5-S-67.8 

-25 

10. 

l-r-,0025 

15.  10004-.5 

20.  79864-3.75 

21.  1.6875  j 

26.  789.7-r 

■1000                 31. 

604.56^-1000 

22.  134.25-; 

-7.5 

27.  1.6875- 

4-6.75                32. 

1220.6744-19 

23.  .045  6 -f- 

.04 

28.  2.0005- 

4-7.24                33. 

144.6955-4-8.5 

24,  733.264 

-33 

29.  128.1754-7.5                34. 

12.345 -4-.00015 

25.  1139-4-9250 

30.  7.024-r 

-2.0005              35. 

15.63386-4-4.367 

36.  549.90254-2.345 

37.  994.8015-h22.33 

38.  600.26234-66.77 

39.  7.006652-4-1.234 

40.  1220.6744-64.246 


41.  .00134094-.583 

42.  .0000026-*-. 004 

43.  5941.86234-66.77 

44.  37.873565-4-8.765 

45.  .0897688 4- ..0202 


46.  245.86776454-405 

47.  20.34407403-i-.21 

48.  12345.432 14-11 1.11 

49.  1.33709774^-.  Ill  11 

50.  72.01440072-4-8.0008 


Applications. — l.  The  circumference  of  a  circle  is  3.14  times 
the  length  of  the  diameter.  Find  the  diameter  of  a  circle  whose 
circumference  is  51.339  yd.  ? 

2.  The  area  of  a  rectangle  is  3414.012  □  yd.,  its  width  is 
125.7  yd.     What  is  its  length  ? 

3.  56.325  cwt.  of  certain  goods  cost  $49.45335;  what  is  the 
cost  of  1  cwt.  ?     Of  1  pound  ? 

4.  36.35  yd.  of  cloth  cost  $117.95;  what  does  1  yd.  cost  at 
the  same  rate  ? 


186  STANDARD  ARITHMETIC. 

Reducing  Common  Fractions  to  Decimals  and  Decimals 
to  Common  Fractions. 

Exercises  on  Diagram,  page  11  U. 

Express  in  decimals  and  also  in  lowest  terms  of  common  frac- 
tions the  parts  of  the  diagram 

1.  In  2,  3,  4,  etc.,  long  slips. 

2.  In  8,  25,  32,  20,  75  small  squares. 

3.  In  2,  8,  14,  25,  125,  175  short  slips. 

4.  In  8,  16,  32,  125,  1875,  625,  3125  tiny  squares. 

183.  Changing  Decimals  to  Common  Fractions. — 5.  Express 
.6  in  the  lowest  terms  of  a  common  fraction. 

Process. — .  6  =  6/10  =  3/5. 

6.  Express  .4,  .8,  .16,  .72,  .75,  .375,  .875,  .4375,  .04,  .0016 
in  lowest  terms  of  common  fractions. 

Note. — The  learner  will  be  able  to  write  out  his  own  rule  for  the  foregoing 
process. 

7.  Express  in  integers  and  common  fractions  1.2,  15.25,  8.6. 

8.  Express  .4%  in  the  lowest  terms  of  a  common  fraction. 

Process.—.  4%  =  -^  =  %0  =  7/15.      (See  Art.  156.) 

9.  In  like  manner  find  the  equivalents  of  .3%,  .23%,  .79/i6, 
,324y7  in  common  fractions. 

184.  Changing  Common  Fractions  to  Decimals. — Any  frac- 
tional part  of  an  object  must  contain  a  like  part  of  the  decimal 
divisions  of  the  object. 

Thus,  1/2  the  diagram,  page  174,  contains  1/2  of  ten  long  slips  =  5  long  slips 
=  .5  ;  50  small  squares  =  50  hundredths,  etc.,  etc. 

a/4  of  the  diagram  contains  1/4t  of  the  decimal  divisions,  as :  2  1/2  long  slips 
=  .21/2  ;  or,  25  small  squares  =.25  of  the  diagram. 

3/8  of  the  diagram  contains  3/8  of  the  long  slips.  3/8  of  10  long  slips  ss  3  3/4 
long  slips  =  .3  3/4.  3/8  of  100  small  squares  =  37  1/i  small  squares  =  .37  V2  I  an^ 
3/8  of  1000  short  slips  =  375  short  slips  =  .375  of  the  diagram. 


DECIMAL  FEAGTIGNS.  187 

185.  Hence,  to  convert  a  common  into  a  decimal  fraction,  we 
take  such  part  of  the  decimal  divisions  of  the  unit  as  is  indicated 
by  the  common  fraction. 

10.  Find  decimals  equivalent  to  the  common  fractions,  %,  3/5, 

3/   9/    11/    12/    21/    8/     27/    101/     333/ 
/8>   /16>   /32>   /25>   /32>   /125?   /64>    /125>    /625« 

11.  Write  in  integers  and  decimals  equivalents  for  3y2,  563%, 
5%,  7%,  9%,  16%. 

12.  Find  equivalents  for  %  %,  %,  %,  %,  5/12,  %4,  16/2l,  *%, 
3  7s,  75/6,  8%,  42/7,  813/14,  9%  in  pure  or  mixed  decimals. 

Suggestion. — The  question  should  be  raised  here,  why  it  is  that  in  Examples 
10  and  11  all  the  common  fractions  are  exactly  reducible  to  decimals,  while  those  in 
12  are  not.  Thus  the  learner  may  discover  for  himself  the  condition  under  which 
exact  decimal  results  are  possible. 

186.  Bule,—1.  To  reduce  common  fractions  to  decimals,  annex 
ciphers  to  the  numerator  of  the  common  fraction,  divide  by  the 
denominator.  Continue  the  process  till  the  division  is  complete, 
or  until  the  result  is  sufficiently  exact. 

2.  Point  off  as  many  decimal  places  in  the  quotient  as  there 
are  decimal  ciphers  annexed  to  the  numerator  of  the  common 
fraction.  If  there  he  not  so  many  places,  ciphers  must  be  pre- 
fixed to  the  significant  figures  to  supply  the  deficiency. 

Note. — The  further  the  division  is  carried,  the  more  exact  is  the  result.  In 
most  cases  sufficient  accuracy  is  reached  in  the  third  or  fourth  place  of  decimals. 

Repetends. — In  the  process  of  division,  if  a  remainder  is  re- 
peated, the  figures  of  the  quotient  will  be  repeated  in  the  same 
order  as  after  its  first  occurrence. 

187.  A  figure  or  set  of  figures  thus  repeated  is  called  a 
Repeating  or  Circulating  Decimal,  or  simply  a  Repetend. 

188.  The  sign  of  a  repetend  is  a  dot  (•)  written  over  the  re- 
peating figure,  or  a  dot  over  the  first  and  last  figure,  if  it  contains 
more  than  one. 

Note.— At  this  point  the  pupil  needs  to  learn  no  more  of  this  subject  than  how 
to  indicate  a  repetend  when  it  occurs,  and  that  he  may  discontinue  the  work  of  divis- 
ion on  the  first  recurrence  of  any  particular  remainder.     (See  Appendix.) 

Examples. — 1-7.  Reduce  the  following  common  fractions  and 
indicate  the  repetends :  %  %  %,  %  %*  8/15,  %. 


188  STANDARD  ARITHMETIC. 

SLATE     EXERCISES. 

Express  equivalents  in  pure  and  mixed  decimals  : 

1.  9.7%,  7.7%,  1.6%*  5.  .0000%  X  .9%. 

2.  $28%,  $17.07%,  .05331/32.         6.  10.111%  X  .033. 

3.  15  %0,  9.60%,  105. 00%.  7.  5.009  X  .08%. 
ool/       171/ 

4'   4V?      7    '  2°-03/5*  8'  108V4  X  %  of  9^' 

9.  Find  the  sum  of  %  and  .54;  the  difference  of  %  and  .54; 
the  product  of  %  and  .54;  the  quotient  of  %  divided  by  .54. 

Find  the  sums  of 

10.  4%,  524.2%,  6.2%,  7,  and  .573%. 

11.  3%  miles,    5%  miles,   4.7  miles,    7.11  miles,  and  99.9% 
miles. 

12.  4.79  lb.,  9%  lb.,  109/20  lb.,  38.59%  lb.,  141.1  lb. 

13.  .125  rod,  .1875  rod,  %  rod,  .5%6  rod,  1.8%  rod. 

14.  1927.961%5  acres,  .00%5  a.,  50.267  a.,  1.709  a. 
Find  the  differences  between 

15.  1.79%  and  .777%;   11.111%  and  11.110%6. 

16.  1.001%  and  10.100%;   7.9753%  and  6.428104%. 


17.  What  number  divided  by  1.25  will  give  the  product  11  X 
1.1X.001%0? 

18.  What  was  paid  for  100  bbls.  flour,  each  196  lb.,  at  $6.66% 
per  100  lb.  ?    For  100  bbls.  pork,  each  200  lb.,  at  $.08%  a  pound  ? 

19.  How  many  wagon  loads  in  a  freight  car  containing  2%6 
tons  sheet  copper,  3.75  tons  sheet  lead,  '57s  tons  sheet  iron, 
7.9375  tons  tin  plate,  1%  tons  being  a  wagon  load  ? 

20.  From  a  sheet  of  lead  weighing  1560.625  lb.,  circular  discs 
were  cut,  weighing,  respective^,  13%  lb.,  17%  lb.,  98.875  lb., 
59.625  lb.,  137%6  lb.,  122%2  lb.,  121%  lb.  What  was  the 
weight  of  the  remnants  (scraps)  ? 


DECIMAL  FRACTIONS. 


189 


189.  To  find  cost  when  number  and  price  per  hundred  or  thou- 
sand  are  given. 

Per  C  is  used  for  per  hundred  and  per  M  for  per  thousand.     (See  page  18.) 

Example. — l.  What   is   the  cost  of  480  lemons 
at  $3.60  a  hundred? 


Written  Work. 
$3.60 
4.80 


28800 
1440 
$17.2800 


Written  Work. 
$7.35 
17.3 


Explanation. — In  480  there  are  4  hundred 
and  80  hundredths  of  a  hundred ;  therefore  we 
find  4  and  80  hundredths  times  the  price  of  1 
hundred. 

Note. — Ciphers  at  the  right  of  a  multiplicand 
or  multiplier  may  be  omitted  in  computation,  in- 
asmuch as  they  do  not  affect  the  value  of  the 
result.    Hence  the  work  may  stand  as  at  the  right. 


$3.6 
4^ 

288 
144 

$17.28 


2205 
5145 
735 
$127,155 


2.  What  must  be  paid  for  17300  bricks  at  $7.35 
per  M  ? 

Explanation.  — 17300  =  17.3  thousand;   hence,  to  find  the 
cost,  at  $7.35  per  thousand,  we  multiply  $7.35  by  17.3. 

3.  Find  the  cost  of  7854  railroad  ties  at  $95.50  a 
thousand. 

4.  Find  the  cost  of  1478  feet  of  lumber  at  $45  per  M. 

5.  Mr.  Smith  bought  50000  shingles  at  70^  a  bundle  of  250, 
and  38750  ft.  of  pine  flooring  at  $18. 75  a  thousand.  What  did 
they  cost  ? 

6.  We  need  45350  bricks;  the  price  being  $6.90  a  thousand, 
how  much  will  they  cost  ? 

7.  Mr.  Wick  bought  280  melons  at  $7.40  a  hundred.  What 
did  they  cost  him  ? 

8.  Find  the  cost  of  2750  laths  at  45^  per  C  ;  of  1950  pick- 
ets at  $12  per  M. 

9.  What  is  the  cost  of  1500  ft.  of  copper  wire  at  $2.85  per  hun- 
dred yards  ? 

10.  How  much  will  the  steel  rails  necessary  to  lay  one  mile  of 

road  cost  at  the  rate  of  $49.30  for  100  ft.  of  rail  ?  (5280  ft.  =  1  mile.) 
9  13 


190  STANDARD  ARITHMETIC. 

Mule.—  Find  the  number  of  hundreds  by  pointing  off  two  fig- 
ures, and  of  thousands  by  pointing  off  three  figures,  on  the  right 
of  the  given  number  (representing  the  quantity),  and  by  this  mul- 
tiply the  price  per  hundred  or  thousand,  as  the  case  may  be. 


190.  To  find  the  cost  when  the  number  of  pounds  and  the  price 
per  ton  (2000  lb.)  are  given. 

Example. — l.  What  will  a  load  of  hay  weighing 

Written    U/n-I/  ° 

2Y2 w  2386  Pounds  cost  at  l19'75  Per  ton  ? 

— Explanation. — There  are  2  thousand  and  386  thousandths  of  a 

l.lyo  thousand  pounds  in  the  load,  and  one  half  as  many,   or   1.193, 

19.75  times  2000  pounds,  or  tons.     Hence  the  value  of  the  hay  is  1.193 

5955  times  $19.75,  the  price  of  1  ton. 

8351  11-14.  Find  the  cost  of 

10737  3500  lb.  of  hay  at  $16  a  ton. 

1193  4835  lb.  of  salt  at  $25  a  ton. 


23.56175  9350  lb.  of  silver  ore  at  $43.50  a  ton. 

'  380  lb.  of  straw  at  $9  a  ton. 

15.  Find  total  freight  charges  on  machinery  shipped  from  New 
York  to  Buffalo  in  the  following  quantities,  @  7/8<p  a  ton  per 
mile  (see  table,  page  48) : 

15000  lb.  locomotive  castings.      81750  lb.  flour-mill  machinery. 

17570  lb.  pumping  machinery.     49975  lb.  saw-  and  planing-mill  machinery. 

16.  What  is  the  cost  of  47.77  tons  of  iron  rails  @  $29%  a  ton. 
What  would  be  the  freight  charges  from  Cleveland  to  Buffalo  @ 
1  %f*  Per  ton  for  a  mile  ? 

17.  Find  cost  of  978  tons  Bessemer  steel  rails  @  $40.33y3, 
freight  being  1  %^  per  ton  for  a  mile,  ordered  in  Cleveland  and 
delivered  in  Jacksonville  ?    (For  distance,  see  page  48.) 

18.  What  did  I  pay  for  2975  pineapples  at  $11.87V2  Per  c  ? 

19.  What  is  the  value  of  9775  lb.  ice  at  $6.75  a  ton  ? 

20.  How  many  thousand   cartridges  can  be  bought  for  $855, 

there  being  5000  in  case,  the  cost  of  a  case  being  $47.50  ? 

Mule.  —  Multiply  the  price  per  ton  by  one  half  of  the  number 
of  thousands  of  pounds  (number  of  tons). 


DECIMAL  FRACTIONS.  191 

Miscellaneous    Examples. 

1.  A  bricklayer  earned  $121.22  in  29  days;  how  much  in  1 
day? 

2.  38  bales  of  cotton  cost  $3213.28  ;  what  is  the  cost  of  1  bale  ? 

3.  The  area  of  a  garden  in  the  form  of  a  rectangle  is  4133.64 
sq.  yd.,  its  length  is  76  yd.      How  wide  is  the  garden  ?    {To find 

the  area  we  multiply  the  length  by  the  width.) 

4.  158  logs  measure  3105.648   feet.     What  do  they  average? 

5.  What  is  the  87th  part  of  53.244  gallons  ?    Of  53.244  qt.  ? 

6.  Divide  %  of  8.236  by  .138  of  %. 

7.  What  decimal  of  2%  yd.  is  %  yd.? 

8.  What  part  of  3  miles  is  %  of  a  mile  ?      (Express  the  answer 
in  decimals.) 

9.  Bought  3. 75  cwt.  beef  at  $.  125  per  lb. ;  find  the  total  cost. 

10.  What  number  multiplied  by  12  will  produce  .1728  ? 

11.  Divide  the  average  of  3.079,  4.276,  5.60554  by  .006. 

12.  If  I  walk  3. 789  miles  an  hour,  how  far  will  my  friend  walk 
in  5  hours,  if  he  walks  only  4/5  as  fast  as  I  do  ? 

13.  The  French  standard  of  measure,  the  meter,  is  39.37  inches 
long  ;  how  many  meters  in  1132.134  yards  ? 

14.  How  much  carpet  1.5  yd.  wide  will  cover  a  floor  22.5  by 

19. 5  ft.  ?     (How  many  widths,  the  carpet  being  laid  from  end  to  end  of  the  room  ?) 

15.  A  regiment  of  550  men  has  on  its  sick-list  .02  of  the  num- 
ber.    How  many  men  are  fit  for  service  ? 

16.  What  decimal  fraction,  multiplied  by  %  of  7%  gives  % 

of  %of  y8? 

17.  The  difference  between  two  numbers  is  17*%n  ;  the  greater 
number  is  25  y9,  what  is  the  smaller  number  ?     (Answer  in  decimals.) 

18.  A  bar  of  iron  8  inches  square  and  1  foot  long  weighs 
216.336  lb.,  what  is  the  total  weight  of  5  pieces  respectively  3,  4, 

5,  6,  and  7  ft.  long  ?      (Only  one  multiplication  necessary.) 


192  STANDARD  ARITHMETIC. 

19.  What  must  the  dividend  be  if  the  divisor  is  38.125  and  the 
quotient  5.25  ? 

20.  What  number  must  be  multiplied  by  7  to  make  %  ?    To 

make  11/12  ?     To  make  2 1/2  ?     (Solve  each  by  decimals.) 

21.  Find  the  price  of  an  ounce  avoirdupois,  if  a  lb.  costs 
$.176;  $.2475. 

22.  The  water  that  will  fill  a  can  which  is  exactly  one  foot 
long,  wide,  and  deep  is  997.7  oz.,  or  62.356  pounds;  hammered 
silver  is  10.511  times  heavier  than  an  equal  bulk  of  water.  What 
is  the  weight  of  21/3  cubic  feet  of  the  silver  ? 

23.  After  selling  78.38  acres  of  his  land,  a  farmer  had 
198.6%  acres  remaining.     How  many  acres  did  he  have  at  first  ? 

24.  Increase  */4  of  7.2  by  %  of  6.5,  and  subtract  from  the 
sum  the  product  of  5  X  1.14. 

25.  Mr.  Smith  has  loaned  $62848  to  different  parties  for  $4% 
per  year  for  every  hundred.  What  is  his  income  per  year  from 
these  loans  ? 

26.  If  one  vd.  of  calico  is  sold  for  $.08,  how  many  yd.  can  be 
had  for  $6%/$4%,  $5%,  17%,  $9%? 

27.  An  agent  collected  $347.35,  and  received  for  the  service  5<f> 
on  every  dollar  collected.     How  much  did  he  get  ? 

28.  In  a  city  of  240768  inhabitants,  it  was  found  that  .125  of 
the  number  could  not  read,  and  only  .875  of  those  able  to  read 
could  write.  How  many  were  there  who  could  not  read  ?  Who 
could  not  write  ? 

29.  How  much  must  be  paid  for  the  use  of  $750  per  year  at 
$5%  a  hundred  ?     For  %  %  %  %  %  %,  %  year?    (Express 

results  in  decimals.) 

30.  The  use  of  $750  cost  me  $37.50  a  year.  What  did  I  pay 
per  hundred  ? 

31.  The  use  of  $1200  for  10  years  cost  Mr.  Lund  $630.  What 
did  the  use  of  $100  cost  him  per  year  ? 


DECIMAL  FRACTIONS.  193 

32.  Mr.  Smith  paid  $45  a  year  at  $4.50  per  hundred  for  the 
use  of  a  certain  sum.     What  was  that  sum  ? 

33.  Mr.  Cain  borrowed  a  sum  of  money  at  $3.25  a  hundred  per 
year,  and  in  5  years  paid  $162.50  for  the  use  of  it.  How  great 
a  sum  was  it  ? 

34.  Find  the  cost  of  6  gal.  3  qt.  vinegar,  at  $.  125  a  gal.     (3  qt. 

=  what  part  of  a  gal.  ?) 

35.  Find  the  cost  of  16  gross  6  doz.  lead-pencils,  at  55^  a  doz. 
(A  gross  is  12  dozen.) 

36.  Find  the  cost  of  IS1/*  yd.  ribbon,  at  $.2325  a  yd. 

37.  Find  the  cost  of  6.25  doz.  cabbage-heads  at  3<fi  apiece. 

38.  Find  the  cost  of  4  gross  10.5  doz.  eggs,  at  t,38*/|  a  dozen. 

39.  If  a  railroad  train  runs  27.125  miles  an  hour,  in  what 
time  will  it  run  303. 6  miles  ? 

Note. — The  distance  around  a  circle  (the  circumference)  is  very  nearly  3.1416 
times  the  distance  across  it  through  the  center  (the  diameter).  Let  the  pupil  care- 
fully measure  the  distance  around  and  across  a  bushel  or  peck  measure,  or  around 
and  across  a  wagon  wheel,  or  any  other  circle,  and  see  if  the  distance  around  is  not 
about  3  */7  times  the  distance  across  it.     (x/7  is  very  little  greater  than  .1416.) 

40.  Find  the  circumference  of  a  circle,  the  di- 
ameter of  which  is  4.2  yards. 

41.  The  diameter  of  a  wagon  wheel  is  40  in. 
How  many  yards  will  the  wheel  progress  in  turn- 
ing 150  times  ? 

42.  Find  the  circumference  of  a  circle,  if  the  length  of  a  ra- 
dius is  3.75  in.     If  the  diameter  is  91.5  in.     (Radius  =  */,  Diameter.) 

43.  A  block  of  gold  measuring  1  in.  long,  wide,  and  thick  (a 
cubic  in.)  weighs  .7003  of  a  lb.,  how  much  does  a  cubic  foot 
weigh  ?     (See  note,  page  103.) 

44.  If  I  add  the  product  of  11.111  by  22.02  to  33.033,  and 
from  this  sum  subtract  277.69721,  what  will  the  remainder  be  ? 

45.  If  you  subtract  the  product  of  5  by  31.565  from  the  sum 
of  the  two  products  15  X  .178  and  50.05  X  3.1,  what  will  remain  ? 


194  STANDARD  ARITHMETIC. 

Bills  and  Accounts. 

191.  An  Account  is  a  record  which  may  include  services  ren- 
dered, goods  sold,  or  money  paid  by  one  person  to  another. 

192.  A  Debtor  is  a  person  from  whom  a  debt  is  due ;  a 
Creditor  is  a  person  to  whom  a  debt  is  due. 

193.  A  Bill  is  the  creditor's  written  statement  of  the  items 
in  his  account  with  the  debtor. 

(94.  Each  item  charged  is  called  a  debit;  each  item  acknowl- 
edged as  received  is  called  a  credit. 

195.  The  Balance  of  an  account  is  the  difference  between  the 
footing  of  the  debits  and  the  footing  of  the  credits. 

196.  To  receipt  a  bill  is  to  write  the  creditor's  name  on  the 
bill  under  the  words  " Received  payment"  or  "Paid." 

A  bill  can  be  receipted  only  by  the  creditor  or  by  a  person  authorized  by  him. 
In  the  latter  case,  the  person  receipting  should  write  under  the  creditor's  name 
"  by  "  or  "  per,"  followed  by  his  own  name  or  initials.  When  a  bill  is  paid  by  a 
promissory  note  or  a  due-bill,  the  fact  may  be  stated  after  the  words  "  Received 
payment." 

197.  To  extend  the  items  of  an  account  is  to  write  in  the 
dollar  and  cent  columns  the  cost  of  each  article  named  at  the 
price  specified.  To  foot  the  several  items  is  to  write  their  sum 
at  the  bottom. 

198.  An  Invoice  is  a  detailed  statement  of  the  quantity,  price, 
and  description  of  goods  sent  to  a  purchaser  or  agent  at  one  time. 
It  includes  also  all  charges,  as  for  packing,  cartage,  insurance,  etc. 

The  following  signs  and  abbreviations  are  commonly  used  in 
business  : 

qfe,  account.  Co.,  company.  Inst.,  this  month. 

Acc't,  account.  C.  O.  D.,  collect  on  delivery.  Int.,  interest. 

Ara't,  amount.  Cr.,  credit,  creditor.  Mdse.,  merchandise. 

@,  at.  Do.  or  ",  the  same.  Pay't,  payment. 

Bal.,  balance.  Dr.,  debit,  debtor.  P'd,  paid. 

Bo't,  bought.  Fr't,  freight. 


DECIMAL  FRACTIONS. 


195 


Let  the  following  bills  be  neatly  and  carefully  copied,  extended, 
and  footed,  with  pen  and  ink.  They  may  also  be  used  as  materials 
for  dictation  exercises. 

1.  New  York,  March  31,  1885. 

Mr.  George  N.  Bell, 

Bought  of  Prince  &  Morton. 


Feb. 

7 

5-Z/jj  $.  #««er                                       @  $.50 

u 

u 

2  doz.  Eggs                                             "     .35 

u 

a 

1  gal.  Molasses 

76 

a 

U 

2  lb.  Mixed  Coffee                                  "     .&? 

M 

#i 

#5  lb.  Lump  Sugar                                "     .07 

u 

a 

i/g  bu.  Potatoes                                      u  1.50 

Mar. 

£ 

3  lb.  Cheese                                              M     .IS 

u 

5 

2  lb.  Raisins                                           u     .15 

$ 

Rec'd  Pay't, 

Prince  &  Morton. 

2. 


Mr.  James  S.  Coolet, 

To  Edward  Willis,  Br. 


Chicago,  Aug.  1,  188^. 


July 

7 

To  1  Cheese-Dish 

u 

u 

M  ■*/#  *&«•  Butter-Plates 

@  $i.50 

a 

if 

"   2-*/<g  <&>s.  Dinner- Plates 

"     £.50 

u 

5 

14  5  Candlesticks 

"       .#5 

u 

itf 

"  5  Pitchers 

4i       .75 

a 

M 

"   1  •*/#  <&>2.  (7w/?s  «?i<Z  Saucers 

"    i.#5 

a 

(« 

"   1%  dos.  i^rfo 

"    4.50 

a 

a 

"   1 1/%>  doz.  Knives 

Rec'd  Pay't, 

"    4.75 

Edward 

Willis. 

50 


196 


STANDARD  ARITHMETIC. 


3. 

1884. 

Cleveland,  Jan. 

Mk.   J.    P.    KlNGSLEY, 

BoH  of  Adam  Johnson. 

11,  188 

5. 

Nov. 

u 
it 
U 

u 

8 
tc 

5 
7 

a 

9  yd.  Cashmere                                    %  %.75 

XU  vd-  Velvet                        "  Jf-50 

6  yd.  Lawn                                           "     .1*"% 
1*1  2  yd.  Silesia                                    "     JO 
%yd.  Cashmere                                "     .65 

Kec'd  Pay't, 

Adam  Johnson, 

i?y  W.  Wright. 

$ 

4.  Knoxville,  Tenn.,  Aug.  15,  1885. 

Me.  George  Cttetiss, 
1885.  In  aceH  with  James  Akden  and  Company. 


July 

# 

« 

a 

tt 

u 

u 

u 

u 

6 

u 

it 

July 

i5 

a 

a 

a 

a 

Dr. 

r<?  i  Plow 

u  2  Extra  Plowshares 

"  life  lb.  Powder 

"  5  lb.  Shot 

"  3  Hoes 

"  5  Rales 


By  20  lb.  Butter 
"   10  bu.  Potatoes 


Ce. 


@  %1.50 
"       .55 

"        .90 
"       .25 


@     $.15 

Balance, 

Rec'd  pay't, 

James  Arden  &  Co. 


$15 


00 


DECIMAL  FRACTIONS.  197 

Atlanta,  Ga.,  Apr.  14,  1882. 
Mb.  George  Eeade, 

To  J.  V.  Camp,  Dr. 


Apr. 

6 

Repairing  House  as  per  contract 

%50 

It 

7 

life  hours'1  labor 

@  %.32lfe 

II 

ii 

12  ft.  Clear  Lumber 

u    .04 

a 

ti 

2  lb.  Nails 

"     .05 

u 

5 

10  hours'*  labor 

"     .8211  2 

U 

(i 

11  ft.  Lumber 

"     .031/2 

II 

ll 

Cartage 

25 

11 

ii 

3  lb.  Nails 

"     .05 

$ 

Paid, 


J.  V.  Camp. 


6. 

Omaha,  Neb.,  Sep.  3,  1882. 

Me.  Geoege  Huelbubt, 

1882. 

ifo  account  with  "Wm.  Powees. 

Aug. 

17 

7b  16  yd.  Frieze  with  Inlay 

@  $.25 

a 

ii 

"  Hanging  14  yd.  Border 

"     .00 

a 

ii 

"   11  Rolls  Paper 

"     .37 

14 

u 

"   Hanging  11  Rolls  Paper 

"     .25 

11 

19 

"   Painting  Kitchen 

$13 

00 

U 

ti 

"  2Qlfeyd.  Painting 

u     .20 

(1 

21 

"  #7  y<Z.  Painting  in  Hall 

"     .&? 

II 

u 

"   i#  yd.  Paper 

"     .12 

M 

ii 

"   6  Aows  Kalsomining 

"     .50 

II 

ii 

"  #  y^.  F^6Z  P<7/>«r 

"      .40 

% 

Rec'd  Pay't  by  note  at  30  d., 

Wm.  Powees. 


198  STANDARD  ARITHMETIC. 

Rule  paper  in  proper  form,  and  make  out  bills  for  the  fol- 
lowing transactions  : 

7.  Mrs.  Cole  bought  of  E.  P.  Dale,  of  Boston,  Feb.  5,  1884,  2 
cans  of  String  Beans,  @  100;  %  bu.  Potatoes,  @  $1.00;  2  lb. 
Tea,  @  600  ;  Feb.  9,  1  lb.  Crackers,  200  ;  1  doz.  Eggs,  320  ;  7  lb. 
Graham  Flour,  @  40 ;  Feb.  16,  3  cans  Tomatoes,  @  120 ;  4  lb. 
Prunes,  @  160 ;  1  doz.  Oranges,  500.  Receipt  the  bill  as  clerk 
for  Mr.  Dale. 

8.  Charles  Martin  bought  of  Joseph  A.  Snow,  of  Pittsburg, 
Pa.,  Feb.  2,  1884,  2%  lb.  Mutton  Chops,  @  220;  %  pk.  Ap- 
ples, @  400 ;  6  lb.  Beef,  @  200 ;  Feb.  6,  %  pk.  Sweet  Potatoes, 
@  300 ;  2  bunches  Lettuce,  @  120 ;  2  qt.  Turnips,  @  50 ; 
Chicken,  4%  lb.,  @  200;  Feb.  9,  2  lb.  Steak,  @  250;  %  pk. 
Apples,  @  700;  1  qt.  Onions,  100;  Feb.  16,  7%  lb.  Beef,  @ 
200  ;  2  cans  of  Peas,  @  180 ;  %  doz.  Oranges,  @  500 ;  Feb.  20, 
9Vt  lb.  Ham,  @  180;  Feb.  26,  2%  lb.  Lamb  Chops,  @  220;  1 
doz.  Oranges,  500.  Mr.  Snow  had  bought  of  Mr.  Martin  3  pt. 
of  Cream,  @  120  a  qt.,  daily  through  the  month.  Make  out  a 
receipted  bill,  using  Bill  4  (page  196)  as  a  model. 

9.  Alfred  E.  Robie  bought  of  John  Turner,  of  New  Haven, 
Conn.,  April  2,  1885,  21/2  lb.  Sausage,  @  140;  %  doz.  Lemons, 
@  250;  2  lb.  Dried  Apples,  @  100;  Apr.  4,  3y4  lb.  Veal 
Chops,  @  200 ;  Apr.  9,  %  pk.  Spinach,  @  700  ;  2  lb.  Mutton, 
@  140 ;  Apr.  14,  x/f  pk.  Apples,  @  700 ;  2  qt.  Sweet  Potatoes, 
@  100 ;  Apr.  18,  63/4  lb.  Beef,  @  200 ;  y2  doz.  Bananas,  @  400 ; 
2  doz.  Pickles,  @  70 ;  2  qt.  Bermuda  Onions,  @  200 ;  Apr.  23, 
3V2  lb.  Steak,  @  220  ;  Apr.  28,  2  lb.  Rhubarb,  @  100  ;  3  bunches 
Radishes,  @  70. 

10.  Mrs.  James  Bird  bought  of  John  Burns,  of  New  Orleans, 
La.,  the  following  articles  :  Feb.  17,  1883,  %  doz.  Linen  Nap- 
kins, @  $1.75;  2*/4  doz.  Damask  Towels,  @  $4.50;  3  Bath 
Towels,  @  $2.40  a  doz.;  Feb.  21,  1883,  2  Table-cloths,  @  $5.50; 
1  Piano-cover,  @  $5.00;  7  yd.  Cambric,  @  $.12y2;  2  pr.  Lace 
Curtains,  @  $2.50  a  pair. 


DECIMAL  FRACTIONS.  199 

11.  Kobert  M.  Miles  bought  of  Lane  &  Bowers,  of  Philadel- 
phia, Pa.,  Nov.  21,  1885,  1  Suit  for  $28 ;  3  Shirts,  @  $1.25  ;  1 
pr»  Shoes,  $5.50  ;  6  pr.  Socks,  @  35^  ;  1  Umbrella,  $2.50  ;  2  pr. 
Gloves,  @  $1. 75  ;  and  4  pr.  Cuffs,  @  35^.  Payment  was  made 
by  note  at  3  months. 

12.  Mr.  George  Ross  bought  of  Kobert  James,  of  Albany, 
N.  Y.,  on  Mar.  13,  1881,  60  yd.  Brussels  Carpet,  @  $.85  ;  40 
yd.  Moquette  Carpet,  @  $1.55  ;  35  yd.  Canton  Matting,  @  $.55  ; 
3  Curtain-poles,  @  $4.50  ;  3  pr.  Nottingham  Lace  Curtains,  @ 
$5.50. 

13.  Albert  Halsted,  in  Qjc  with  George  Eeese  :  Aug.  7,  1881, 
1%  days'  work,  @  $3.25  ;  44  ft.  Pine  Lumber,  @t  1.06%;  1  lb. 
Nails,  $.07;  work  on  Bookcases  as  per  contract,  $13.00;  65  ft. 
Pine  Lumber,  @  $.06%;  %  lb.  Nails,  @  $.07.  Cr.  by  cash, 
$5.00. 

14.  Mr.  Robert  Holden,  of  Brooklyn,  New  York,  bo't  of  Stan- 
ley, White  &  Co.,  of  New  York  city/ Mar.  11,  1884,  3  doz.  8  in. 
Thermometers  on  polished  walnut,  @  $10 ;  1%  doz,  8  in.  Parlor 
Thermometers,  @  $4  apiece ;  5  doz.  tin-case  Thermometers,  @ 
$5  ;  9  Aneroid  Barometers,  @  $5  ;  15  pr.  Opera-glasses,  @  $4.25  ; 
3  Microscopes,  @  $15  ;  1  large  first-class  Microscope,  $350 ;  2 
Amateur  Photographic  Cameras,  @  $25.     Paid  by  note  at  3  mo. 

15.  Mrs.  H.  R.  Otis  bo't  of  Richard  Hayes,  June  11,  1880, 
1  pr.  Ladies'  Kid  Button  Shoes,  $6;  June  13,  2  pr.  Ladies'  Patent- 
Leather  Oxford  Ties,  @  $4.50  ;  1  pr.  Misses'  Kid  Button  Shoes, 
$3.50  ;  2  pr.  Infants'  Black  Kid  Button  Shoes,  soft  soles,  @  $.45  ; 
1  pr.  Child's  Pebble  Spring-heel  Button  Shoes,  $2. 

16.  James  R.  Baldwin  bo't  of  Robert  Price,  Dec.  19,  1884,  1 
copy  "Little  Men,"  $1.35;  1  "Modern  Explorers,"  $10;  1 
"Three  Vassar  Girls  in  South  America,"  $1.30;  1  "Rose  in 
Bloom"  and  "Eight  Cousins,"  $2  ;  13  vol.  Shakespeare,  @  $1 ; 
3  vol.  "Diamond  Edition  Poetry,"  @  $.90;  5  vol.  "Companion 
Edition  Poetry,"  @  $1.25  ;  6  vol.  Hawthorne,  @  $1.3o  ;  1  vol. 
"Sports  and  Pastimes  for  American  Boys,"  $1.25. 


200  STANDARD  ARITHMETIC. 

Suggestions  for  Original  Problems. 

1.  Pupils  will  find  suggestions  for  original  problems  in  the 
Miscellaneous  Exercises ;  or,  it  may  be  required  that  they  con- 
struct problems  of  their  own  after  models  dictated  by  the  teacher. 

2.  Having  obtained  reliable  information  from  parents  and 
others  in  regard  to  prices,  trade  customs,  etc.,  they  can  make  out 
bills,  and  furnish  items  for  bills  to  be  made  by  the  class. 

3.  They  may  draw  diagrams  showing  the  forms  and  dimen- 
sions of  lots  to  be  fenced,  dictate  the  kinds  of  fences  to  be  built, 
prices  of  boards,  posts,  labor,  nails,  etc.,  and  require  the  whole 
cost.  They  may  give,  in  like  manner,  the  information  necessary 
to  reckon  the  cost  of  digging  cellars,  building  walls,  laying  board, 
stone,  or  brick  walks,  etc.,  etc.  Pupils  may  often  obtain  from 
each  other  such  information  as  may  be  needed. 

4.  Let  illustrations,  like  the  one  on  page  174,  be  required, 
showing  .33,  1.27,  etc.,  etc.,  of  given  squares. 

5.  Let  pupils  obtain  where  they  can,  the  data  necessary  to 
enable  them  to  calculate  the  cost  of  papering,  carpeting,  plaster- 
ing, the  schoolroom. 

6.  Pupils  who  have  a  little  constructive  skill  may  make  paper 
boxes,  and  require  their  classmates  to  calculate  their  contents — 
how  many  quarts  of  blackberries  or  vinegar  they  will  contain,  etc. 

7.  Try  the  experiment  of  ascertaining  the  height  of  some  tall 
tree  or  steeple,  by  measuring  the  length  of  its  shadow,  and  the 
length  of  the  shadow  cast  at  the  same  moment  by  a  stick  or  post, 
the  length  of  which  above  ground  can  be  easily  measured. 

8.  Give  the  dimensions  of  a  pile  or  load  of  wood,  and  ask,  How 
many  cords  ?  or  of  a  wood-shed,  and  ask,  How  many  cords  can 
be  piled  in  it  ?  or  the  length  of  a  pile  of  wood,  and  ask  how  high 
it  must  be  to  contain  some  required  number  of  cords. 

9.  Give  the  dimensions  of  a  box  containing  a  gross  of  such 
crayons  as  are  used  at  the  blackboard,  and  ask  the  length  and 
width  of  a  case  which  will  exactly  contain  a  gross  of  such  boxes. 


CHAPTER   XI. 

MEASURES. 

(99.  The  length,  breadth,  and  height  of  objects  are  their 
dimensions. 

A  line  has  only  one  dimension — length. 
A  surface  has  two  dimensions — length  and  breadth. 

A  solid  or  space  has  three  dimensions — length,  breadth,  and  height  or 
thickness.  

Measures  of  Extension. 

200.  Measures  used  to  ascertain  how  long  a  line  is,  or  in 
calculating  the  size  (extent)  of  a  surface  or  solid,  are  called 
Measures  of  Extension.  These  are  the  Linear,  Square,  and 
Cubic  Measures. 

Linear  or  Line   Measure. 

201.  In  measuring  length  or  distance,  linear  or  line  measure 
is  used.     The  standard  unit  is  the  yard. 

Table. 
12  Inches  (in.)  =  1  Foot  (ft.). 
3  Feet  =  1  Yard  (yd.). 

16V2  Feet       )      «--,-, 

(or5V2yards)r  =  1Rod(rd-)- 
320  Rods  =  1  Mile  (mi.). 

Equivalents. 
1  mile  =  320  rods  =  1760  yards  =  5280  feet  =  63360  inches. 

Notes. — 1.  For  measuring  cloth  the  yard  is  divided  into  halves,  fourths,  eighths, 
and  sixteenths.    In  the  United  States  custom-houses  it  is  divided  decimally. 

2.  A  Furlong  =1/8  mile. — The  rod  is  also  called  a  Pole  or  Perch. 

3.  A  Pace  is  variously  estimated  from  3  to  3.3  feet. 

4.  A  Line  =]L  inch. 


202  STANDARD  ARITHMETIC. 

202.  The  mile  given  in  the  table  is  the  mile  used  in  land 
measurements.  Its  length  is  fixed  by  law,  and  is  called  the 
statute  mile.  It  is  thus  distinguished  from  the  geographical 
mile  of  the  following  table,  used  on  shipboard  and  at  sea. 

Table. 

6  Feet  =  1  Fathom. 

120  Fathoms  s=  1  Cable  Length. 

1.15  +  Common  Miles         =  1  Geographical  or  Nautical  Mile. 

3  Geographical  Miles  )       .  _ 

«  M ~      ox  f  =  1  League  (at  sea). 

or  3.45  +  Statute   "     )  &  J 

A  Knot  corresponds  to  one  geographical  or  nautical  mile,  and  is  used  to  esti- 
mate the  speed  of  vessels  at  sea. 

Note. — In  the  absence  of  a  more  exact  instrument  the  hand  was  formerly  used 
as  a  measure.  From  this  we  have  the  Palm  (breadth  of  four  fingers)  =  about  3 
inches ;  the  Hand  (the  breadth  of  palm  and  thumb,  used  in  measuring  the  height  of 
horses  at  the  shoulder)  =  4  inches ;  the  Span  (the  distance  between  the  tips  of  the 
thumb  and  the  little  finger,  when  the  hand  is  extended  against  a  flat  surface)  = 
about  9  inches,  or  1/4  of  a  yard. 

ORAL     EXERCISES. 

How  many 

1.  Feet  in  3%,  4%,  7%,  4.4,  11%,  33%  yd.? 

2.  Feet  in  25,  16,  30,  39,  14%  in.  ? 

3.  Yards  in  1%1?  2%,  5,  8%i  rods.? 

4.  Rods  in  %  %,  %  mi. ;  in  121,  49%  yd.  ? 

5.  Inches  in  1%,  6%,  3%,  5%,  7%2  ft.? 

6.  Feet  in  2%,  3%,  10%,  6%  fathoms? 

Surveyors'   Measure. 

203.  Gunter's  Chain,  used  in  measuring  roads  and  the  bound- 
ary lines  of  land,  is  4  rods  (=  66  ft.)  in  length.  It  has  100  links, 
each  7.92  inches  long. 

Table. 
7.92  Inches  =  1  Link  (li.). 
100  Links  m  1  Chain  (ch.). 
80  Chains  =  1  Mile  (mi.). 


MEASURES. 


203 


5 

Va  yards. 

1 

CI 

Square  or  Surface   Measure. 

2  04-.  There  is  no  measure  which  is  di- 
rectly applied  to  a  surface  to  find  its  extent. 
Even  if  there  were  such  a  measure,  it  would 
be  difficult  to  apply  it.  Suppose,  for  instance, 
that  we  wished  to  ascertain  how  many  square 
yards  there  arc  in  a  plot  of  ground  5  1/2  yards 
long  and  5  lft  yards  wide.  If  we  had  a  square- 
yard  measure  we  might  perhaps  mark  off  25 
square  yards  and  the  fractions  of  a  yard,  as  in 
the  diagram.  But  it  would  be  much  easier  to 
measure  the  length  and  breadth  with  a  yard- 
stick, and  then  compute  the  number  of  square 
yards  in  the  surface. 

205.  The  square  inch,  foot,  yard,  rod,  and  mile  are  derived 
from  corresponding  linear  measure. 

Table. 
144  sq.  Inches  =  1  sq.  Foot.  3074  sq.  Yards  =  1  sq.  Rod. 

9  sq.  Feet     =  1  sq.  Yard.  160  sq.  Rods       =  1  Acre. 

640  Acres  ==  1  sq.  Mile  (or  Section  of  Land). 

Equivalents- 
d  mile,    acres.        d  rods.  □  yards.  n  feet.  d  inches. 

1  =  640  =  102400  =  309  7600  =  2  78  78400  =  4014489600 

The  sign   □   is  used  for  the  abbreviation  "  sq."    In  written  exercises,  either 
can  be  used. 

Note. — The  acre  has  no  corresponding  denomination  in  linear  measure.     A 
square,  measuring  208.'71  +  feet  on  each  side,  contains  1  acre. 


ORAL     EXERCISES. 

How  many 

1.  Square  yards  in  12,  1881,  26,  100,  66  □  ft.  ? 

2.  Acres  in  Vie,  %  %  %   D  mi- ? 

3.  Square  feet  in  a  board  6  ft.  6  in.  long,  2/13  ft.  wide  ? 

4.  A  board  18  in.  long  contains  half  a  d  ft.;  how  wide  is  it  ? 

5.  How  many  □  rods  in  %,  %,  8/8,  3/ie  of  an  acre  ? 

6.  How  many  acres  in  a  half  section  of  land  ?    In  a  quarter  ? 


204 


STANDARD  ARITHMETIC. 


/ 

/ 

Cubic   Measure. 

206.  To  measure  a  block  of  marble,  or  to  find  how  much  a  box,  a  bin,  or  a 
room  will  contain,  we  have  to  ascertain  its  length,  breadth,  and  height  or  thickness, 
by  a  linear  measure,  as  a  foot-rule,  a  yard-stick,  or  a  tape-line ;  and,  with  the  aid  of 
the  dimensions  thus  found,  to  calculate  the  contents  of  the  block,  or  bin,  or  room, 
in  Cubic  Measure,  that  is,  we  calculate  how  many  times  the  room,  or  the  space 
occupied  by  the  block,  would  contain  some  known  cubic  unit,  such  as  a  cubic  inch, 
cubic  foot,  etc. 

207.  A  Rectangular  Solid  is  a  solid  haying  six  rectangular 
faces. 

208.  A  Cube  is  a  rectangular  solid  having 
six  equal  square  faces.     (See  also  page  103.) 

The  figure  at  the  left  represents  the  outlines  of  a  cubic 
foot,  with  a  layer  or  course  of  cubic  inches  at  the  bottom. 
With  this  figure  before  the  pupil  let  him  answer  the  fol- 
lowing questions  :  1.  How  many  cubic  inches  in  the  course 
represented?  2.  How  many  such  courses  are  needed  to 
complete  the  foot  ?  3.  How  many  cubic  inches  in  a  cubic 
foot?  In  l/12?  3/4?y24?etc. 
On  inspection  of  the  figure  at  the  right,  an- 
swer the  following  questions :  1.  How  many  cubic 
feet  in  a  cubic  yard?  2.  What  is  the  length 
of  each  edge  of  a  cubic  foot  ?  3.  Can  you  lift  a 
cubic  foot  of  granite  ? 

4.  How  many  cubic  feet  in  a/3  of  a  cubic 
yard?  5.  How  many  cubic  feet  in  */9  of  a  cubic 
yard  ? 

6.  How  many  cubic  inches  in  a  cubic  foot  ? 
7.  How  many  cubic  inches  in  a  cubic  yard  ?  In 
*/3  of  a  cubic  yard? 

209.  Thus  the  cubic  inch,  foot,  and  yard  are  derived  from 
the  corresponding  linear  measures. 

Table. 
1728  Cubic  Inches  =  1  Cubic  Foot. 
27  Cubic  Feet     =  1  Cubic  Yard. 

Equivalents. 
1  Cubic  Yard  =  27  Cubic  Feet  =  46656  Cubic  Inches. 

Note. — Higher  denominations  than  these  are  seldom  referred  to. 


MEASURES.  205 

Wood   Measure. 

210.  Wood  cut  in  "lengths"  of  4  feet  is  called  "cord  wood." 
A  pile  of  cord  wood  four  feet  high  and  eight  feet  long,  or  equal 
bulk  of  other  material,  is  called  a  Cord. 

211.  One  foot  in  length  of  such  a  pile  is  called  a  cord  foot. 

Table. 
16  Cubic  Feet  =  1  Cord  Foot. 

8  Cord  Ft.  or  128  Cubic  Ft.  =  1  Cord. 


ORAL     EXERCISES. 

How  many 

1.  Cubic  feet  in  %  2%  1%,  3%  cu.  yd.  ? 

2.  Cubic  feet  in  %  %  %  2.625  cords  ? 

3.  Cubic  inches  in  an  iron  bar  13 y2  in.  long,  3y3  in.  wide, 
y2  in.  thick  ? 

4.  Cubic  inches  in  a  brick  8  by  4  by  21/2  inches  ? 

5.  Cubic  yards  in  a  wall  6  ft.  high,  9  in.  thick,  and  20  yd. 

long  ?     (6  ft.  =  2  yd.,  9  in.  =  */4  yd.) 

6.  Cord  feet  in  3%,  7.125,  4.375  cords? 


Measures  of  Capacity. 

212.  For  measuring  fruits,  berries,  roots,  grains,  and  other 
dry  commodities,  we  use  Dry  Measure.  The  standard  unit  is 
the  Bushel  =  2150.42  cubic  inches. 

Dry  Measure. 

Table. 
2  Pints  (yt.)  =  1  Quart  (qt.). 
8  Quarts       =  1  Peck  (pk.). 
4  Pecks         =  1  Bushel  (bu.). 

Fquivalents. 
1  Bushel  =  4  Pecks  =  32  Quarts  =  64  Pints. 

Charcoal  and  coke  are  frequently  measured  by  the  chaldron,  of  36  bushels. 


206  STANDARD  ARITHMETIC. 

213.  For  measuring  liquids,  such  as  water,  wine,  vinegar, 
milk,  etc.,  we  use  Liquid  Measure.  The  standard  unit  is  the 
Gallon.  =  231  cubic  inches. 

Liquid   Measure. 

Table. 
4  Gills  (si.)  =  1  Pint  (pt.). 
2  Pints        =  1  Quart  (qt.). 
4  Quarts      =  1  Gallon  (gal.). 

Equivalents. 
1  Gallon  =  4  Quarts  =  8  Pints  =s  32  Gills. 

2 1 4-.   Comparison  of  Dry  and  Liquid  Measures. 
The  Dry  Quart  contains        67.2    cubic  inches. 
The  Liquid  Quart  contains  57.75  cubic  inches. 

Notes. — 1.  Thus  it  will  be  seen  that  the  retailer  who  uses  the  liquid  instead  of 
the  dry  quart,  in  measuring  berries  and  small  fruits,  cheats  his  customers  out  of  a 
little  more  than  one  quart  in  seven. 

2.  Barrels,  tierces,  hogsheads,  puncheons,  pipes,  butts,  tuns,  etc.,  have  no  stand- 
ard capacity.  The  quantity  of  liquid  contained  in  them  is  usually  found  by  actual 
measurement,  called  gauging. 

3.  When  the  barrel  is  spoken  of  as  a  measure  of  the  capacity  of  vats,  cisterns, 
etc.,  3172  gallons  are  meant.  In  measuring  beer,  the  barrel  has  36  gallons,  and 
V/z  barrels  (or  54  gal.)  make  a  hogshead. 


ORAL     EXERCISES. 

How  many 

1.  Quarts  in  2%  3%,  6%,  4.25,  5.5  gal.? 

2.  Pints  in  2,  10,  15,  180  gi.? 

3.  Gallons  in  9,  14,  27,  17,  30,  111,  63  pt.  ? 

4.  Quarts  in  7,  20,  31,  15,  50,  25,  35,  45  gi.? 

5.  Quarts  in  %  2%,  l7/8  bu.? 

6.  Pints  in  31/*  8%,  6.25,  9%,  10.125  qt.? 

7.  Bushels  in  10,  1.6,  23,  2.8,  17  pk.? 

8.  Pecks  in  %%  4%,  3%,  5.75  bu.  ? 

9.  Quarts  in  5,  8%  10 %  33  y3,  13.825  pt.? 


MEASURES.  207 

Measures  of  Weight. 

215.  For  weighing  gold,  silver,  the  precious  stones,  etc.,  Troy 
Weight  is  used.  The  standard,  unit  is  the  Troy  pound  =  5760 
grains. 

Troy  Weight, 

Table. 

24  Grains  (gr.)      =  1  Pennyweight  (pwt.). 

20  Pennyweights  =  1  Ounce  (oz.). 

12  Ounces  =  1  Pound  (lb.). 

Equivalents. 

1  Pound  =  12  Ounces  ==  240  Pennyweights  =  5760  Grains. 

Practical  illustrations  of  Troy  weight  are  to  be  found  in  the  United  States  coins : 
The  gold  dollar  weighs  25.8  grains ;  the  silver  dollar,  41 2^2  grains ;  the  small  silver 
coins,  385.8  grains  to  a  dollar  (that  is,  10  single  dimes,  or  4  quarters,  or  2  half- 
dollars,  weigh  385.8  grains).  The  nickel  5^  piece  weighs  77.16  grains ;  the  3^  piece, 
30  grains,  and  the  bronze  1^  piece,  48  grains. 

Gold  and  silver  are  bought  and  sold  by  the  ounce,  weights  of  these  metals 
never  being  expressed  in  pounds.  The  carat,  very  nearly  equal  to  3x/5  Troy 
grains,  is  used  in  weighing  diamonds  and  other  precious  stones. 

The  word  carat  is  also  used  in  expressing  the  number  of  parts  of  pure  gold  in 
articles  of  jewelry,  etc.  If  18  parts  out  of  24  are  pure  gold,  and  the  remaining  6 
parts  are  alloy,  the  metal  is  said  to  be  of  18  carats,  etc. 


How  many  0RAL   exercises. 

1.  Pennyweights  in  3%,  5.3,  6%,  9.2,  4%  oz.? 

a.  Ounces  in  1%,  3%,  4%,  7%  lb.? 

3.  Pounds  in  16,  30,  27,  9,  9y3,  23.3  oz.? 

4.  Grains  in  %  %  1%  pwt.?    In  1.5,  2%  oz.? 

5.  Ounces  in  45,  56,  90,  18,  50  pwt.? 

6.  Ounces  of  pure  gold  in  44  oz.  of  watch-cases,  18  carats 
fine?  

7.  Of  how  many  carats  is  a  mixture  of  27  oz.  gold  and  13  % 
oz.  alloy  ? 

8.  How  many  pwt.  of  alloy  must  be  put  with  25  pwt.  of  pure 
gold  to  make  a  mixture  of  20  carats  ? 


208  STANDARD  ARITHMETIC. 

Apothecaries'  Weight. 
216.  Apothecaries'  Weight  is  used  only  by  physicians  in  pre- 
scribing and  by  apothecaries  in  compounding  medicines.     When 
sold  by  weight,  avoirdupois  weight  is  used. 

Table. 

20  Grains  (gr.)  =  1  Scruple  3. 

3  Scruples       =  1  Dram   3  . 

8  Drams  =  1  Ounce  | . 

12  Ounces  =  1  Pound  ft. 

Equivalents. 

&  1  =  l  12  =  3  90  =  3  288  =  gr.  5760. 

Note. — 1.  It  should  be  observed  that  in  this  weight  the  signs  precede  the  num- 
bers to  which  they  belong.  2.  The  grain,  the  ounce,  and  the  pound  are  of  the  same 
value  as  the  corresponding  denominations  in  Troy  weight. 


u  _  ORALEXERCISES. 

How  many 

1.  Pounds  in  ?  36  ?    I  40  ?    \  27  ?    I  75  ? 

2.  Grains  in  3  I1/,  ?  3  %  ?  3  %  ?  3  3.75  ?  3  .1  ? 

Avoirdupois  Weight. 

217.  For  the  common  purposes  of  trade,  Avoirdupois  Weight 
is  used.     The  standard  unit  is  the  pound  of  7000  grains. 

Table. 
16  Ounces  (oz.)         =  1  Pound  (lb.). 
100  Pounds  =  1  Hundredweight  (cwt). 

20  Hundredweight  =  1  Ton  (T.). 
The  term  cental  is  beginning  to  be  used  for  hundredweight. 

Equivalents. 
1  Ton  =  20  Hundredweight  =  2000  Pounds  =  32000  Ounces. 

Formerly  112  lb.  were  reckoned  a  hundredweight,  and  2240  lb.  a  ton.  This 
weight  is  still  used  in  weighing  iron,  coal  at  the  mines,  ores,  and  goods  on  which 
duties  are  paid  at  the  United  States  custom-houses. 

218.  Comparison  of  Troy  with  Avoirduoois  Weight. 
Avoirdupois:  1  lb.  =  7000  grains.        1  oz.  =  437y2  grains. 
Troy:  1  lb.  =  5760  grains.       1  oz.  =  480       grains. 


MEASURES. 


209 


ORAL      EXERCISES. 

How  many 

1.  Ounces  in  1%  2%,  5%,  45/16,  8%,  6%  lb.? 

2.  Pounds  in  %  1%,  7.1,  6%,  61/*,  12  %0  cwt? 

3.  Pounds  in  12,  46,  22,  33.6,  29,  176  oz.? 

4.  Cwt.  in  2%,  3.%,  4%,  11.2,  9.9  T.? 

5.  Pounds  in  1.3,  2%,  3%,  5%,  7%  T.? 

219.  Weight  being  very  commonly  employed  in  estimating 
quantities  of  grains,  roots,  etc.,  the  weight  of  the  bushel,  as  fixed 
by  law  in  many  States,  for  some  of  the  more  important  commodi- 
ties, is  given  below.* 

The  general  usage  is  found  in  the  second  column.  In  the 
third,  exceptions  are  noted  so  far  as  known.  (See  Haswell,  Ed.  1885, 
and  Report  No.  14,  H.  R.,  46th  Congress,  1st  Session.) 


Commodities. 

Lb.  per 
bu. 

Exceptions. 

Barley 

48 

56 
32 

56 
60 

60 
60 

Ariz,  and  Wash.,  45  ;  Cal ,  60 ;  Md.  and  Penn.,  47  ; 
N.  H.  and  Del.,  not  reported. 

Ariz.,  54 ;  Cal.,  52 ;  N.  Y.,  58. 

Iowa,  Mont.,  and  Mo.,  35 ;  Md.,  26 ;  Neb.  and  Ore., 
34 ;  Me.,  N.  H.,  and  N.  J.,  30 ;  Wash.,  36 ;  Ky., 
33x/3  ;  Del.,  not  reported. 

Cal.,  54 ;  La.,  60 ;  Del.  and  Me.,  not  reported. 

Ohio,  58 ;  Ariz.,  Cal.,  Del.,  La.,  Md.,  Penn.,  not  re- 
ported. 

Conn.,  56 ;  R.  I.,  not  reported. 

No  exceptions  reported. 

Shelled  corn 

Oats 

Rye 

Potatoes  

Wheat 

Pease 

Usage  in  regard  to  the  following  articles  is  not  so  uniform  as 
in  case  of  those  given  in  the  foregoing  list : 

Corn  in  the  ear.     Variously  estimated  from  68  to  70  lb. 

Corn  meal.     Del.,  44  lb. ;  111.,  48  lb. ;  most  other  States,  50  lb. 

Beans.    Me.,  64  lb. ;  N.  Y.,  62  lb. ;  many  others,  60  lb. 

Clover  seed.     Mont.,  45  lb. ;  N.  J.,  64  lb. ;  Penn.,  62  lb. ;  in  almost  all  others, 

60  1b. 
Timothy  seed.     Wis.,  46  lb. ;  N.  Y.  and  Mont.,  44  lb. ;  Dakota,  42  lb. ;  in  many 

others,  45  lb. 
Mineral  coal.     Ky.  and  Penn.,  76  lb. ;  Ind.,  70  lb. ;  in  most  others,  80  lb. 


210  STANDARD  ARITHMETIC. 

The  following  standards  are  generally  accepted  : 

100  lb  of  grain  or  flour  =  1  cental.         196  lb.  of  flour  s=  1  barrel. 

100  lb.  of  dry  fish  =  1  quintal.       200  lb.  ot  beef  or  pork  =  1       " 

100  lb.  of  nails  as  1  keg. 


220.  Measures  of  Value. 

United  States  or  Federal   Money. 
For  the  United  States  or  Federal  Money  table,  see  Art.  89. 

The  gold  eoins  are  the  $1,  $f*/a  (quarter-eagle),  $3,  $5  (half-eagle),  $10 
(eagle),  and  $20  (double-eagle)  pieces.  The  silver  coins  are  the  $1,  50^,  250,  and 
100  pieces.  The  50  and  30  pieces  are  made  of  nickel;  the  cent  of  bronze. 
Other  coins  are  occasionally  found  in  circulation,  but  are  no  longer  coined,  such  as 
the  trade-dollar,  the  200,  the  50  and  the  30  silver  pieces,  and  the  20  piece  of 
bronze. 

Canadian   Money. 

22 (.  The  unit  of  Canadian  currency,  like  that  of  the  United 
States,  is  called  a  dollar.  It  is  divided  into  100  cents,  and  the 
cent  into  10  mills. 

The  legal  coins  are  —  Gold :   the   British  sovereign,   worth 

$4.8665,  and  the  British  half-sovereign;  Silver:  the  50^,  25^, 

10^,  and  5^  pieces  ;  Bronze :  1$. 

The  silver  and  bronze  coins  have  the  same  values  as  the  corresponding  coins  of 
the  United  States. 

English  or  Sterling  Money. 

Table. 

4  Farthings  (far.)  =  1  Penny  (d.). 

12  Pence  =  1  Shilling  (s.). 

20  Shillings  =  1  Pound  (£). 

Equivalents. 
1  Pound  =  20  Shilling  =  240  Pence  =  960  Farthings. 

The  coins  of  Great  Britain  are — Gold :  the  sovereign  =  $4.8665,  and  the  half- 
sovereign  ;  Silver:  the  crown  (5  shillings)  =  $1,216  +  ,  and  the  half-crown;  the 
florin  (2  shillings)  =  $.486  ;  the  shilling  =  $.243  ;  the  sixpenny,  fourpenny,  and 
threepenny  pieces;  Copper:  the  penny,  half-penny,  and  the  farthing  (2/4  penny). 
The  guinea  =  21  shillings,  though  no  longer  coined,  is  frequently  mentioned  as  if 
in  common  use. 


MEASURES.  211 

222.   Money  of  Other  Countries. 

a,  French  money :  1  franc  (fr.)  =  100  centimes  (c.)  =  19.3^ 
in  United  States  money. 

The  Gold  coins  of  France  are  the  100,  50,  20,  10,  and  5  franc  pieces ;  the 
Silver  coins  are  the  5,  2,  1,  1/2,  and  1/5  franc  pieces;  the  Bronze,  10,  5,  2,  and 
1  centime  (pronounced  sonteem)  pieces ;  Copper,  10  and  5  centimes.  The  values 
of  these  coins  are  indicated  by  their  relations  to  the  franc. 

b.  German  money:  1  mark  (reiehsmark)  \yX>  m.)  =  100 
pfennigs  (pf.)  =  23.8^  in  U.  S.  money. 

The  Gold  coins  of  the  German  Empire  are  the  20  and  10  mark  pieces ;  the 
Silver  coins  are  the  5,  3,  2,  1,  and  ]  /2  mark  and  20  pfennig  pieces ;  the  Nickel, 
10  and  5  pfennig ;  the  Copper,  2  and  1  pfennig.  The  Thaler  (silver)  =  $.'746, 
and  the  Groschen  (silver)  =  2  1/2$,  are  also  in  common  use. 

For  the  values  of  other  foreign  coins,  see  Appendix. 

223.  The  following  approximations  are  sufficiently  exact  for 
general  estimates :  One  U.  S.  Dollar  may  be  counted  as  equal  to 
5  Francs  (France,  Belgium,  and  Switzerland),  or  to  5  Lire  (Italy), 
or  to  5  Peseta  (Spain),  or  to  4  Shillings  (England),  or  to  4  Marks 
(Germany). 

ORAL    EXERCISES. 

How  many 

1.  Pence  in  2,  3%,  9%,  14%,  21  s.?    In  %  crown  ? 

2.  Farthings  in  8%,  I*%»  23%,  3  pence? 

3.  Pounds  in  50,  15,  75,  105,  130,  244  shillings  ? 

4.  Shillings  in  30,  6,  45,  33,  81,  108  pence  ? 

5.  Shillings  in  £7%?  £15%?  £22%?  22/3  guineas? 

6.  Centimes  in  %,  1%,  5%,  17%,  28%  francs? 

7.  Marks  in  175,  210,  1728,  3042  pfennigs  ? 

8.  Dollars  may  be  counted  as  equal  to  18  roubles  ?  42  roubles  ? 
166  roubles  ? 

9.  Dollars  may  be  counted  as  equal  to  34,  78,  92,  118  s.  ?  To 
1  guinea?    To  75,  130,  195  peseta? 

10.  Dollars  may  be  counted  as  equal  to  250000  francs  ?  To 
£340000  ?     To  3000  marks  ?     To  1500  roubles  ? 


212  STANDARD  ARITHMETIC. 

Definitions. 

224.  A  point  has  position,  without  length,  breadth,  or  thick- 
ness. 

225.  A  line  is  the  path  of  a  point  in  motion.  If  the  point 
moves  without  change  of  direction,  the  path  is  a  straight  line. 
If  the  point  changes  its  direction  continually  while  moving,  the 
path  is  a  curved  line. 

226.  If  the  moving  point  passes  around  a  fixed  point,  so  that 
its  distance  from  the  fixed  point  does  not  vary,  the  path  of  the 
moving  point  is  the  circumference  of  a  circle,  and  the  fixed 
point  within  is  the  center  of  the  circle. 


227.  For  the  measurement  of  angles  (see  Art.  56),  the  circum- 
ference of  the  circle  is  conceived  to  be  divided 
into  360  equal  parts,  called  degrees.  The  angle 
in  this  figure  is  an  angle  of  90  degrees — one  fourth 
of  the  360  equal  parts  into  which  the  circumfer- 
ence is  supposed  to  be  divided. 

228.  An  angle /of  90  degrees  (written  90°) 
is  a  right  angle.     An  angle  of  less  than  90°  is  an  acute  angle. 
An  angle  greater  than  90°  is  an  obtuse  angle. 

229.  Two  lines  which  form  an  angle  of  90°  are  said  to  be 

perpendicular  to  each  other.     (See  also  Art.  57.) 

Note. — A  degree,  being  1/360  part  of 
any  circumference,  is  very  minute,  if  the 
circle  is  a  small  one;  but  a  degree  of  the 
circumference  of  the  earth  is  69.16  miles  in 
length.  A  degree  of  the  sun's  circumfer- 
ence is  about  7444  miles  long.  Compare 
these  with  one  degree  on  the  protractor,  as 
here  represented.  A  protractor  is  an  instru- 
ment used  for  the  measurement  of  angles. 

230.  For  more  exact  measurements,  the  degree  (°)  is  divided 
into  minutes  ('),  and  the  minutes  into  seconds  ("),  according  to 
the  following  table  : 


MEASURES.  213 

Circular  Measure, 

Table. 
60  Seconds  ( ")  =  1  Minute  ('). 
60  Minutes       =  1  Degree  (°). 
360  Degrees       =  1  Circumference. 


ORAL     EXERCISES. 

How  many 

1.  Degrees  in  %  %  */«,  %  V»  Vs*  circumference  ? 

2.  Minutes  in  3%°  ?   2%°  ?   5%°  ?  4%°  ?  7%°  ? 

3.  Degrees  in  180'  ?   150'  ?   3600'  ?   420"  ?   1800"  ?   6000"  ? 

4.  Minutes  in  900"  ?   720"  ?   342"  ?   1275"  ?  3333"  ? 


231.   Time  Measure. 

Table. 

60  Seconds  (sec.)  =  1  Minute  (min.). 

60  Minutes  =  1  Hour  (h.). 

24  Hours  =  1  Day  (d.). 

7  Days  =  1  Week  (wk.). 

365  Days  5  Hours  48  Min.  46.4  Sec.  =  1  Solar  Year  (yr.). 

The  year  given  in  the  table,  which  is  a  little  less  than  365  l/4r  days,  is  the  time 
it  takes  the  earth  to  go  around  the  sun,  and  hence  the  time  required  for  a  complete 
change  of  seasons.  But  to  count  this  quarter  of  a  day  with  every  year  would  be 
extremely  inconvenient.  It  is  much  easier  to  count  one  additional  day  every  fourth 
year,  and  hence  this  is  generally  done,  but  not  always,  for  in  a  hundred  years  we 
should  thus  gain  nearly  a  day  too  much ;  so  the  hundredth  years  (centennial  years) 
are  commonly  counted  as  ordinary  years ;  but  here  again  we  have  to  say  not  always, 
for  we  should  thus  lose  nearly  a  day  in  400  years.  Hence  the  centennial  years 
divisible  by  400  are  counted  as  leap  years. 

The  following  is  the  rule  by  which  leap  years  may  be  known 
for  several  thousands  of  years  to  come  : 

232.  All  years  divisible  by  4,  except  centennial  years  not 
divisible  by  400,  are  leap  years. 

233.  There  are  12  months  in  a  year.  The  number  of  days  in 
each  is  given  in  the  following 


10 


214 


STANDARD  ARITHMETIC. 


Table. 

Months.  Days. 

1st.   January    (Jan.) 31 

2d.    February  (Feb.) 28  or  29 

3d.    March       (Mar.) 31 

4th.  April        (Apr.) 30 

5th.  May  (May) 31 

6th.  June         (June) 30 


Months.  Days. 

7th.  July  (July) 31 

8th.  August        (Aug.) 31 

9th.  September  (Sept.)   ...   30 

10th.  October      (Oct.) 31 

11th.  November  (Nov.) 30 

12th.  December   (Dec.)  ....  31 


The  29th  day  of  February  is  the  day  added  to  make  a  leap  year. 

The  following  lines  are  used  to  aid  the  memory  in  recalling  the  number  of  days 
in  the  several  months  : 

"  Thirty  days  hath  September, 
April,  June,  and  November ; 
All  the  rest  have  thirty-one, 
Except  February  alone, 
Which  has  but  28  in  fine, 
Till  leap  year  gives  it  29." 

The  long  months  may  be  distinguished  by  observing  that  their  names  are  the 
only  ones  that  contain  the  letter  c,  or  that  have  a  for  their  second  letter,  and,  except 
June,  the  only  ones  that  have  u  for  their  second  letter.     (See  whether  this  is  true.) 


ORAL     EXERCISES. 

How  many 

1.  Days  in  2%,  4%,  14  wk.? 

2.  Hours  in  %%  4%,  3%  d.?  1  wk.? 

3.  Minutes  in  1%,  8%,  2%,  43/5,  12,  24  h.? 

4.  Weeks  in  9,  30,  23,  17,  60,  90,  365  d.? 

5.  Days  in  Aug.?  Apr.?   Dec.?  Jan.?    Sept.?    Feb.?  July? 
Nov.?  Mar.?  Oct.?  June?  May? 

6.  Days  in  the  year  1886?  1894?  1896?  1900?  1800?  2000? 


234-.    Miscellaneous  Measures. 


12  Units 

12  Dozen 

20  Units 

5  Score 


Counting. 

=  1  Dozen  (doz.). 
=  1  Gross  (gro.). 
=  1  Score. 
=  1  Hundred. 


24  Sheets 
20  Quires 
2  Reams 
5  Bundles 


Paper. 

=  1  Quire  (qu.)t 
=  1  Ream  (r.). 
as  1  Bundle. 
=  1  Bale. 


MEASURES. 


215 


1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 


MISCELLANEOUS  ORA 

How  many 

Inches  in  2  %  3.75  ft.?  21. 

□  ft.  in3Va,  5.75  □  yd.  ?  22. 
Ou.  in.  iniy8,  1.5  cu.  ft?  23. 
Pints  in  11.75,  9%  qt.  ?  24. 
Pecks  in  3%,  5.125  bu.?  25. 
Quarts  in  15,  20.5  pk.  ?  26. 
Grains  in  7,  7.66%  pwt.?  27. 
Seconds  in  3y4,  5y8  min.  ?  28. 
Minutes  in  .125,  5/6  h.  ?  29. 
Degrees  in  .162/3  circum.  ?  30. 
Dozen  in  3y4,  4.375  gross?  31. 
Pwt.  in  11.5,  9%oz.?  32. 
D  ft.  iniy8,  2.5  □  yd.?  33. 

□  in.  in  .375,  l3/8  □  ft.  34. 
Cu.  ft.  in  1.88 yt  OIL  yd.?  35. 
Feet  in  ?%  7. 5  fathoms?  36. 
Feet  in  3  y4  cable  lengths  ?  37. 
Yards  in  1.5,  2.6  rd.?  38. 
Rods  in  7.7,  11.5  yd.?  39. 
Feet  in  9.6,  15.6  in.?  40. 


L    EXERCISES. 

How  many 

Lb.  pork  in  2  %  3.7  bl? 
Pence  in  145/6  s.  ? 
Pounds  in  32  oz.  avoir.  ? 
Pounds  in  8.4  oz.  Troy  ? 
Gallons  in  2.6,  17  qt.? 
Gills  in  27,  1.3  pt.? 
Ounces  in  36,  96  pwt.  ? 
Lb.  Troy  in  3,  18  oz. 
□  ft.  in  288,  72  p  in.  ? 
Feet  in  10.8,  8.4  in.? 
Feet  in  7.2,  10%  in.? 
Cu.  yd.  in  10.8  cu.  ft.? 
Bushels  in  23,  97  pk.  ? 
Pecks  in  23,  97  qt.  ? 
Pints  in  7.2,  39  gills  ? 
Fath.  in  iy8  cable  lengths  ? 
Feet  in  15  %  14  hands  ? 
Quarts  in  73,  95  pt.  ? 
Pecks  in  3.2,  .96  qt.? 
Lb.  pork  in  5,  iy4bl.? 


Definitions. 

Compound  Denominate  Numbers. 

235.  Number  when  applied  to  specified  objects  is  said  to  be 
concrete. 

236.  Number  when  not  applied  to  specified  objects  is  said 
to  be  abstract.  ______ 

237.  To  measure  a  quantity  is  to  find  how  many  times  it 
contains  some  known  quantity  used  as  a  standard  of  comparison. 


216  STANDARD  ARITHMETIC. 

238.  A  known  quantity,  fixed  by  law  or  custom  as  a  standard 
of  comparison,  is  called  a  Unit  of  Measure. 

Note. — A  yard  is  a  standard  fixed  by  law  for  measuring  length  or  distance.  A 
hand  is  a  standard  fixed  by  custom  for  estimating  the  height  of  horses. 

239.  Units  of  measure  have  special  denominations  or  names 
by  which  they  are  designated,  and  hence  they  are  called  Denom- 
inate Units.      (Denomination  means  name.) 

240.  A  Denominate  Number  is  a  number  of  denominate 
units. 

241.  A  Simple  Denominate  Number  is  one  that  consists  of 
units  of  only  one  denomination. 

242.  A  Compound  Denominate  Number  is  one  that  consists 
of  units  of  two  or  more  denominations. 

243.  Changing  the  denomination  in  which  a  quantity  is  ex- 
pressed is  called  Reduction. 

244.  Reduction  Ascending  is  changing  an  expression  of 
quantity  from  a  less  to  a  greater  unit  of  measure. 

245.  Reduction  Descending  is  changing  an  expression  of 
quantity  from  a  greater  to  a  less  unit  of  measure. 


SLAT  E     EXERCISES. 

Example. — l.  Eeduce  3  gal.  2  qt.  1  pt.  3  gi.  to  gills. 

Analysis. — Since  there  are  4  qt. 

Process,  in  one  gallon,  there  must  be  3  times 

3  gal.  2  qt.  1  pt.  3  gi.  4  qt.  =  12  qt.  in  3  gal. ;    12  qt.  + 

^  2  qt.  =  14  qt.      Since  there   are  2 

pt.   in    1   qt.,    there    must    be    14 


14  qt. 


times    2  pt.  =  28  pt.  in  14  qt. ;  28 


2  r         Caution.  —  Say  ,        pt.  +  1    pt.  =  29   pt.      Since   there 

2g     ^  \  12,  14 ;  not  4  times  /        are  4  gi.  in  1  pt.,  there  must  be  29 

*  1  3  =  12,    and    2    are  L        times  4  gi.  in  29  pt.  =  116  gi. ;  116 

^  '  14,  etc.  /        gi.  +  3  gi.  =  119  gi.      Hence,  in  3 

119  gi.  gal.    2    qt.    1    pt.    3   gi.    there    are 

119  gi. 


MEASURES. 


217 


Example. — 2.  Keduce  119  gi.  to  higher  denominations. 


Analysis. — Since  there  is  1  pt.  in  4  gills,  there 
arc  as  many  pints  in  119  gi.  as  there  are  times  4  gi. 
=  29  times,  with  3  gills  remaining.  Since  there  is 
1  qt.  in  2  pt.,  there  are  as  many  quarts  in  29  pt.  as 
there  are  times  2  pt.  sr  14  times,  with  1  pt.  remain- 
ing. Since  there  is  1  gal.  in  4  qt.,  there  are  as 
many  gallons  in  14  qt.  as  there  are  times  4  qt.  =  3 
times,  with  2  qt.  remaining.  Hence,  in  119  gi.  there 
are  3  gal.  2  qt.  1  pt.  3  gi. 

Note. — Analysis  here  supersedes  the  necessity  for  any  rule. 

Reduce 


Process. 
4)119  gi. 
2)29  pt.  +  3  gi. 
4)14  qt.  +  1  Pt. 
3  gal.  +  2  qt. 


1.  375.96  inches  to  yards. 

2.  2480  oz.  to  hundredweight. 

3.  23  h.  .48  min.  to  seconds. 

4.  29738.7  inches  to  rods. 

5.  5. 33  y3  days  to  minutes. 

6.  96  oz.  of  lead  to  pounds. 

7.  96  cu.  ft.  to  cu.  yd. 

8.  1  gro.  10  doz.  to  units. 

9.  3  mi.  173  yd.  2  ft.  to  in. 

10.  19.3  pecks  to  bushels. 

11.  5238  far.  to  shillings. 

12.  29  wk.  6  d.  to  hours. 

13.  495  sheets  to  quires. 

14.  69472  lb.  to  hundredweight. 

15.  13  mi.  1.537  yd.  to  yards. 

16.  £6  17  s.  10  d.  to  farthings. 

17.  5620  hours  to  weeks. 

18.  593  yd.  1.8  in.  to  inches. 

19.  27  gro.  11  doz.  to  units. 

20.  25  T.  7  cwt.  to  ounces. 

21.  157  quires  to  reams. 

22.  187.6  quarts  to  pecks. 


23.  5  bu.  3  pk.  7  qt.  to  pints. 

24.  10  wk.  3  d.  10  h.  to  sec. 

25.  27  lb.  9  oz.  Troy  to  grains. 

26.  705  quarts  to  bushels. 

27.  5. 934  feet  to  yards. 

28.  5638  d.  to  pounds  sterling. 

29.  14  h.  36  min.  .5  sec.  to  sec. 

30.  893  units  to  gross. 

31.  14  mi.  18  rods  .6  yd.  to  yd. 

32.  19  cwt.  46  lb.  9  oz.  to  oz. 

33.  3  wk.  5  d.  .19  h.  to  min. 

34.  484  pecks  to  bushels. 

35.  276457  ounces  to  tons. 

36.  4563  shillings  to  pounds. 

37.  654  dozens  to  gro. 

38.  2  s.  6  d.  3  far.  to  farthings, 

39.  8349250  seconds  to  days. 

40.  5  gal.  3.5  qt.  to  pints. 

41.  14  pk.  1  qt.  to  pints. 

42.  628  pints  to  pecks. 

43.  12  bu.  1.75  pk.  to  pints. 

44.  1  mi.  58  yd.  .8  in.  to  in. 


218 


STANDARD  ARITHMETIC. 


Reduce 

45.  945. 6  ounces  to  lb.  avoir. 

46.  25  yr.  79  d.  to  days. 

47.  2572  gills  to  quarts. 

48.  1  mi.  13.62  yd.  2  ft.  to  in. 

49.  2500  inches  to  feet. 

50.  17  cwt.  95  lb.  to  pounds. 

51.  93  reams  2  quires  to  quires. 

52.  976.3  far.  to  pounds. 
53..  42345  ounces  to  tons. 

54.  2  cwt.  75  lb.  to  ounces. 

55.  2  qt.  1  pt.  .3  gi.  to  gills. 

56.  7892.8  minutes  to  days. 

57.  29650  seconds  to  hours. 

58.  19  gr.  8  doz.  10  units  to  units. 

59.  71  bu.  3  pecks  to  quarts. 


60.  593  pints  to  gallons. 

61.  25971  yards  to  miles. 

62.  3000  gills  to  gallons. 

63.  79  tons  2  cwt.  to  pounds. 

64.  930780  minutes  to  weeks. 

65.  738  reams  to  sheets. 

66.  £345  18  s.  8  d.  to  pence. 

67.  7453  sheets  to  reams. 

68.  284  gal.  3  qt.  1  pt.  to  gills. 

69.  7  gr.  6  doz.  11  units  to  units. 

70.  127  T.  15  cwt.  25  lb.  to  lb. 

71.  4  reams  22  sheets  to  sheets. 

72.  9256.35  feet  to  miles. 

73.  13  gal.  1  pt.  2  gi.  to  gills. 

74.  57289  hundredweight  to  T. 


la   How 

Analysis. 

2.  How 

Analysis. 

3.  How 

Analysis 

4.  How 

5.  How 

6.  How 

7.  How 

8.  How 


Reduction  of  Denominate  Fractions. 

Reduction  Descending. 

ORAL     EXERCISES. 

many  ounces  in  7  pounds  avoirdupois  ? 
— 7  lb.  =  7  x  16  ounces,  or  112  ounces. 
many  cents  in '%,  %%  %,  %o,  n/5o,  3/io  dollar  ? 

— $1  =  1000 ;   V,  dol.  =  V,  of  1000  =  500. 

many  inches  in  %  %  %  %  %  %  %  *<>ot  ? 
— 1  foot  =  12  in. ;  2/3  foot  =  2/3  of  12  in.  =  8  in. 

many  inches  in  %  %  %  %  %,    7/10yard? 

many  gills  in      %  %  %  %  %,  %  quart  ? 

many  pints  in    %,  5/6,  %  %,  3/10,  %  gallon  ? 

many  grains  in  %,  %  %  %  %,    %  oz.  Troy  ? 

many  ounces  in  %  %  %,  %  %    %  lb.  Troy? 


MEASURES.  219 

SLA1E     EXERCISES, 

On  pages  216  and  217  the  pupil  learned  to  change  integral  numbers  from  higher 
to  lower  and  from  lower  to  higher  denominations.  Here  he  will  find  the  same  prin- 
ciples applied  to  the  reduction  of  fractional  expressions  from  one  denomination  to 
another. 

Examples. — (l.)  Reduce  3  bushels  to  pints.  (2.)  Reduce  % 
bu.  to  pints.     (3.)  Reduce  .  75  bu.  to  pints. 

These  problems  differ  from  each  other  only  in  this,  that  in  the  first  the  number 
of  bushels  to  be  reduced  is  expressed  by  an  integer,  in  the  second  by  a  common 
fraction,  and  in  the  third  by  a  decimal  fraction.  They  are  all  solved  by  multiplica- 
tion, and  the  reasons  for  multiplication  are  the  same. 

(1.)  3bu.  (2.)  %bu.  (3.)  .75  bu. 

A  J  J 

12  pk.  12/4  =  3pk.  3pk. 

8                          _8  8 

96  qt.                        24  qt.  24  qt. 

2                               2  2 

192  pt.                       48  pt.  48  pt. 

4.  How  many  units  in  2%  gross  ? 

Analysis.— 2  gr.  =  2  x  144  units  =  288  units.  z/3  gr.  =  2/3  of  144  units  =  98 
units.     2S8  +  96  =  384  units. 

5.  How  many  □  inches  in  y4,  y8,  2/5,  2yg,  33/4   n  yards  ? 

6.  How  many  □  feet  in  %  %  iy4,  23/8,  7,  6  a  rods  ? 

7.  How  many  inches  in  $%  33/4,  6%  9%  feet  ? 

8.  How  many  cu.  inches  in  2!/4,  63/7,  55/12,  37/23  cu.  feet  ? 

9.  How  many  pints  in  33/4,  3%  45/7,  63/5  bushels? 

10.  How  many  pennyweights  in  35/8,  57/12,  93/5  pounds  ? 

11.  How  many  feet  in  %  %  %,  1%,  %  %  rods  ? 

12.  How  many  grains  in  7/8,  34/9,  n/12,  57/16  pounds  avoir.  ? 

13.  How  many  hours  in  65/9,  37/8,  27/16,  5n/12  months  ? 

14.  How  many  feet  in  7/8,  4/9,  5/12,  11/16  mile  ? 

15.  How  many  gills  in  %  1%  34/5,  25/48  gallons? 


STANDARD  ARITHMETIC. 
Reduction  Ascending. 

ORAL     EXERCISES. 

1.  What  part  of  a  lb.  is  2,  4,  5,  10,  12,  15  ounces  avoir.  ? 

Analysis. — 2  oz.  are  2/16  of  a  pound  avoirdupois  because  they  are  2  of  the  16 
equal  parts  (ounces)  into  which  a  pound  can  be  divided. 

2.  What  part  of  a  bushel  is  6,  9,  13,  17,  21  quarts  ? 

3.  What  part  of  a  month  is  2  %  days  ? 

21/ 
Analysis.— 1  d.  =  1/30  mo.,  2y2  d.  =  -—^-2  mo.  =  5/eo,  or  J/12  month. 

4.  What  part  of  a  shilling  is  %  penny  ? 


SLATE    EXERCISES. 

Examples. — (l.)  Keduce  1  pt.  to  the  fraction  of  a  bushel.  (2.) 
Eeduce  %  pt.  to  the  fraction  of  a  bushel.  (3.)  Reduce  .2  pt.  to 
the  decimal  of  a  bushel. 

These  are  similar  problems,  and  are  all  solved  by  division,  the  divisors  being 
the  same  in  each  case. 

(1.)  2)1  pt.  (2.)  2)%  pt.  (3.)  2)0.2  pt. 
8)%  qt.                       8)V10  qt.  8)0.1  qt. 

4)%6  pk.  4)%0  pk.  4)0.0125  pk. 

yM  bu.  y320  bu.  0.003125  bu. 

246.  From  the  above  illustrative  examples  it  may  he  seen  that 
the  process  of  reducing  fractions  is  the  same  as  that  of  reducing 
integers  from  one  denomination  to  another. 

4.  How  many  days  in  34%,  37%,  48yl0,  50%  hours? 

This  question  if  given  in  full  would  be,  "  How  many  days  and  what  fraction  of 
a  day  in,"  etc. 

5.  How  many  gallons  in  75%  pints?  In  83%,  92%,  102 "/„ 
quarts?    In  56%,  48%  gills? 

6.  What  part  of  a  hundredweight  is  3/5,  5/8,  7/10,  23/7,  4%, 
7%2,  19%  pounds? 

7.  Reduce  to  acres :  %>0,  1%,  845%,  98374%  □  yd. 


MEASURES.  221 

To  Integers  of  Lower  Denominations. 

SLATE     EXERCISES. 

Example. — l.  Find  the  value  of  the  fractional  part  of  217/36 
lb.  Troy  in  integers  of  lower  denominations. 


Common  Fractions. 

2|"/t.n>.Troy. 
12 

204/    _5|24/ 
20 

;      pWt. 

=8gr. 

Decimal  Fractions. 

2»V,.lb.=2.47»/,lb. 
2|.472/9  lb. 
12 

5.|G62/8  oz. 
20 

480/    _     ,lf/ 
/36       d|     /86 
24 
288/ 

13.|33V8  pwt. 
24 
8gr. 

Answer  to  loth. — 2  lb.  5  oz.  13  pwt.  8  gr. 

2.  How  many  pounds  and  ouncas  in  %,  %,  3/8,  5/9,  7/10,  "/«, 
%n  "As,  9%i  cwt.? 

Analysis.— 2/3  cwt.  =  2/3  of  100  lb.  =  66  2/3  lb. 

2/3  lb.  =  2/3  of  16  oz.  =  l02/3  oz.    Hence  2/3  cwt.  =  66  lb.  102/3  oz, 

3.  How  many  □  yards  and  n  feet  in  3/8,  %,  %,  4/9  acre  ? 

4.  How  many  hours,  minutes,  and  seconds  in  1/21,  4/7,  %,  3/i0, 
Vic  7Ao,  2%7  day  ? 

5.  How  many  feet  and  inches  in  7/8,  4/7,  6/n,  7/18  yard  ? 

6.  How  many  pence  and  farthings  in  %,  7s?  9/io>  "/»  12/i3> 
14/i7,  6%  shillings? 

7.  How  many  weeks,  days,  and  hours  in  23/4,  5%,  11/19,  21/31, 
3/17  common  years  ? 

8.  How  many  cubic  yards  and  feet  in  1%,   95/7,   7yl2,  3y8, 
1113/120  cords  ? 

9.  How  many  pennyweights  and  ounces  in  11  %,  7y8,  52/13, 
67y13,  88 B/»  25yi6  pounds  Troy? 

10.  How  many  grains  and  scruples  in  3  T/g,    3  9y8,    3  3y6, 

3  7y480  ?  in  i  iy18,  i  %  i  y33  ?  in  ib  7y52,  &  iyn,  s>  sy21  ? 


22'2 


STANDARD  ARITHMETIC. 


Change  to  integers 

1.  7s  ton. 

2.  .1125  ton. 

3.  .1325  pint. 

4.  3/9  CWt. 

5.  .125  hhd. 

6.  .99  bl.  pork. 

7.  .38675  oz.  Troy. 

8.  .9975  ft. 

9.  yl2  d  yd. 

10.  .7846  acre. 

11.  4/5  mile. 


SLATE     EXERCISES. 

of  lower  denomination  : 

12.  3/10  day.  23. 

13.  .8  cord  of  wood.  24. 

14.  .98974  qt.  25. 

15.  .7859  cwt.  26. 

16.  .7775  oz.  Troy.  27. 

17.  %,  ton.  28. 

18.  9/n  bl.  of  flour.  29. 

19.  .375  ream.  30. 

20.  y6  cu.  yd.  31. 

21.  y8  cu.  ft.  32. 

22.  3/7  □  mile.  33. 


%  acre. 

Vis  rod. 

%se  lb.  avoir. 

5/6  degree. 

3/5  of  5/7  lb.  avoir. 

3/5  of  1  cwt.  56  lb. 

y7  of  1  mi.  160  rd. 

%  of  %  lb.  Troy. 

.735  bl.  of  beef. 

.9x.87  1b.  Troy. 

.1755  yd. 


To  Fractions  of  Higher  Denominations. 

SLATE     EXERCISES. 

Eeduce  5  d.  14  h.  24  min.  to  a  fraction  of  a  week. 

Explanation. — Reducing  the  two  periods  of  time  to  be  compared,  to  the  same 
denomination,  we  have 

1  wk. 

7 
74. 

24 
168  h. 

60 
10080  min. 

5  d.  14  h.  24  rain,  is or  4/5  of  a  week. 

10080         /5 

247.  Second  Method. — The  lowest  denomination  may  be  re- 
duced to  a  fraction  of  a  unit  of  the  next  higher ;  then  this  frac- 
tion, together  with  the  integer  to  which  it  now  belongs,  may  be 
reduced  to  a  fraction  of  the  next  higher,  and  so  on  till  the  entire 
compound  number  is  reduced  to  the  required  fraction,  as  follows  : 


5  d.  14  h.  24  min. 
_24 
134  h. 

60 
8064  min. 


Analysis. — 5  d.  14  h. 

24  min.  =  8064  min.     One 

week  =  10080  min.     8064 

8064 

of    10080 


mm.   ii 
min 


10080 
Hence, 


MEASURES.  223 

Suggestion. — The  work  is  conveniently  arranged  by  writing  the  several  denom- 
inations in  a  column,  beginning  with  the  lowest,  and  writing  the  resulting  fractional 
quotients  on  the  right  of  a  light  vertical  line,  as  below.  In  the  progress  of  the 
work  this  line  is  disregarded.     It  is  useful  only  to  prevent  mistakes. 

Analysis. — 1  min.  =  1/60h. 

24min.  =  «Veo  =  V«h. 
Pr0C2SS-  2/5  h.  +  14h.  =  142/5  h. 

60)24  min. 


24)14  h. 


7)5  d. 


2^—  lh.=  724d 


14V5h.  =  ^d. 


5  d.  14  h.  24  min.  =  4/5  wk.  14  */, 


72, 


A 20  d.  =  3A  d. 
3/5d.+  5d.  =  53/6d. 


1  d.  =  V7  wk. 

•  ■■/• 

53/5  d.  =  -f  wk. 

5^  Wk.  =  »/„    =  V6    Wk. 

The  operation  by  decimals  is  as  follows  : 

p  Explanation. — Here  the  process  of  reasoning  is  precisely 

fin\9A      '  *^e  same  as  ^or  sim^ar  reductions  in  integers,  thus:  Since  there 

/       min.  .g  j  jj  jn  go  mm ^  there  are  as  many  h.  in  24  min.  as  there  are 

24)14.4  h.  times  60  min.  in  24  min.  =  .4  times,  etc.  etc.     But  this  process 

7)   5.6  d.  differs  from  the  process  of  reduction  in  integers  in  the  addition 

0  o  wfr  of  the  higher  denominations  as  we  come  to  them,  while  in  inte- 

gers there  are  no  such  additions  to  make. 


SLATE     EXERCISES. 

What  part  of 

1.  1  ton  is  7  cwt.  79  lb.  11  oz.  ?       9.  4  □  miles  is  347  acres  ? 

2.  £5  is  1100  d.  ?  10.  59°  is  13°  13'  13"  ? 

3.  1  cwt.  is  79  lb.  9%  oz.?  11.  13  cu.  ft.  is  578  cu.  in.? 

4.  3  acres  is  1700  □  ft.?  12.  3  oz.  is  5%  pwt.  ? 

5.  1  yr.  is  89  d.  1  h.  12  min.?  13.  7  days  is  37  min.  37  sec? 

6.  £1.7835  is  £1  15  s.  8.04  d.?  14.  1  yr.  is  89  d.  17  h.  8  min.? 

7.  6.75  bu.  is  3  pk.  3  qt.  ?  15.  52  d.  16  h.  is  49  d.  9  h.  ? 

8.  5  d  ft.  is  289  d  in.  ?  16.  1  cwt.  is  13  lb.  16  oz.  ? 


224  STANDARD  ARITHMETIC. 

What  part  of 

17.  25  cu.  ft.  is  864  cu.  in.  ?  32.  1  ton  is  6  cwt.  7  lb.  ? 

18.  6  d.  1  far.  is  4  d.  20  h.  ?  33.  %  lb.  Troy  is  12  gr.  ? 

19.  13  cords  is  13  cu.  ft.?  34.  17  h.  is  19  min.  13  sec? 

20.  1.25  ton  is  17%  oz.  ?  35.  17  h.  is  .1175  d.  ? 

21.  13  yd.  is  13  in.?  36.  777  oz.  Troy  is  3  lb.  11  oz.? 

22.  13  gal.  is  3  qu.  1  pt.  1  %  gi.  ?  37.  %  lb.  avoir,  is  21  gr.  ? 

23.  36  gal.  is  27  gal.  2  qt.  1  pt.  ?  38.  %  lb.  avoir,  is  %  lb.  Troy  ? 

24.  1728  cu.  in.  is  445  cu.  in.  ?  39.  3  cwt.  99  lb.  is  %  ton  33  lb.  ? 

25.  1  lb.  Troy  is  7  oz.  6  pwt.  ?  40.  69  cwt.  is  69  lb. 

26.  1  lb.  Troy  is  11  oz.  7  pwt.  ?  41.  2  n  mi.  is  345  a.  17  n  rd.  ? 

27.  1  ton  is  47.73  lb.  ?  42.  5  cords  is  7.125  cord  feet  ? 

28.  3/4  mi.  is  527.3994  yd.  ?  43.  K>  3  is  3  3  3  1  32  gr.  16  ? 

29.  2/3  acre  is  420.1883  n  yd.  ?  44.  180°  is  5°  18'  22"  ? 

30.  1  oz.  Troy  is  1  oz.  avoir.  ?  45.  1  ch.  is  3  rd.  3  li.  5  in.  ? 

31.  1770  oz.  is  y3  cwt.?  46.  1  yr.  is  5  h.  46.4  sec? 


Addition. 

(Compound  Denominate  Numbers.) 

Example. — 1.  What  is  the  sum  of  13  gal.  2  qt.  1  pt.  3  gi.; 
14  gal.  2  qt.  2  gi.;  7  gal.  3  qt.  3  gi.;  9  gal.  1  qt.  1  pt.  2  gi.; 
6  qt.  1  pt.  1  gi.  ? 

Explanation. — Numbers  of  the  same  denomination 
are  written  in  the  same  column  for  convenience  of  ad- 
dition. The  sum  of  the  column  of  gills  is  11  =  2  pt. 
3  gi.  The  3  gi.  is  written  under  the  column  added, 
and  the  2  pt.  are  added  with  the  column  of  pints. 
The  sum  of  the  pints  is  5  =  2  qt.  1  pt.  The  1  pt.  is 
written  under  the  column  of  pints,  and  the  2  qt.  are 
added  to  the  quarts,  and  so  on. 

Jy        Q  3  Note. — The  pupil  will  perceive  that  the  process 

at  the  left  is  like  that  of  simple  addition,  except  that 
instead  of  the  divisor  being  always  10,  as  in  simple  numbers,  it  varies  with  the  de- 
nomination. It  is  always  as  many  units  of  the  denomination  of  the  column  added 
as  are  required  to  make  a  unit  of  the  next  higher. 


Written  Work. 

Gal. 

qt. 

Pt. 

gL 

13 

2 

1 

3 

14 

2 

0 

2 

7 

3 

0 

3 

9 

1 

1 

2 

6 

1 

1 

2. 


MEASURES. 

225 

SLATE 

EXERCISES. 

T. 

cwt. 

lb. 

oz. 

3. 

Mi 

yd. 

ft 

in. 

11 

18 

77 

11 

4 

1678 

2 

11 

32 

11 

31 

10 

2 

1123 

1 

10 

43 

17 

63 

13 

3 

1456 

2 

9 

Ed. 

yd. 

ft. 

in. 

5. 

A. 

d  rd.       d  yd 

□  ft 

d  in. 

5 

3 

2 

8 

2 

115       20 

3 

31 

8 

0 

1 

9 

7 

218       32 

G 

15 

15 

4 

1 

10 

1 

25       31 

8 

25 

10 

1 

2 

3 

3 

34      27 

7 

100 

39 

4 

2 
1 

6 
6 

15 

75      21 

7 
2 

27 
36 

39 

5 

1 

0 

15 

75      22 

0 

63 

es. — 1. 

In  E) 

i,  4,  the  sum 

of  the  ya 

rds  is 

10 

,  or  1  rod  4 

7 

1   3* 

L ;  we  write 

Oz. 

pwt. 

gr. 

44: 

5 

16 

15% 

16 

1 

14 

83% 

15 

0 

17 

0%2 

14 

2 

4 

21% 

20 

3 

19 

8% 

18 

14      12       22^24    83 


the  4,  and  for  the  1/2  yd.  we  add  1  ft.  6  in. 

2.  In  Ex.  5,  the  sum  of  the  a  yards  is 
112,  or  3  □  rods  21 1/4  d  yd. ;  write  the  21 
□  yd.,  and  for  the  '/4  d  yd.  add  2  □  ft. 
36  d  in. 

3.  The  fractions  of  the  lowest  denomi- 
nation being  added  together,  and  reduced, 
the  resulting  integer,  if  any,  is  added  to  the 
given  integers  of  that  denomination. 


Examples.— 7.  Find  the  sum  of  13  cwt.  21  lb.  13  %  oz. ;  3  cwt. 
18  lb.  97/10  oz.;  25  cwt.  31  lb.  15%  oz. 

8.  Add  58  gal.  3  qt.  1  pt.  3%  gi.;  45  gal.  3  qt.  1  pt.  1% 
gi.;  38  gal.  1  qt.  1  pt.  3%  gi.;  26  gal.  3  qt.  3y3  gi. 

9.  Add  £7305  14  s.  8%  d.;  £8737  13  p.  4%  d.;  £513  6  s.  5  d.; 
£67  5  s.  10%  d. 

10.  Add  37  cu.  yd.  15  cu.  ft.  1084  en.  in.;  9  cu.  yd.  13  cu. 
ft.  1556  en.  in.;  86  cu.  yd.  22  cu.  ft.  695  cu.  in.;  24  cu.  yd.  8 
cu.  ft.  924  cu.  in. 

11.  Add  17  tons  11  cwt.  99  lb.  15  oz.;  7  cwt.  97  lb.  13  oz.;  7 
tons  7  cwt.  7  lb.  7%0  oz.;  11  tons  11  cwt.  11  lb.  11  oz.;  179  cwt. 
1780  lb.  11797  oz.  ;  137  tons  19  cwt.  89  lb.  15  oz. 


226  STANDARD  ARITHMETIC. 

Subtraction. 

Example. — l.  Mr.  Jones  had  £4  4  s.  2d.,  out  of  which  he  paid 
Mr.  Smith  £1  3  s.  6  d.      How  much  did  Mr.  Jones  have  left  ? 

Explanation. — To  pay  the  6  d.  Mr.  Jones  obtains  change  Written  Work. 

(12  d.)  for  a  shilling,  and  putting  this  with  the  2  d.  he  has  £        s.         d. 

14  d.     14  d.  —  6  d.  =  8  d.     Having  taken  1  s.  from  4  s.  there  4        4        2 

remain  but  3  shillings,  which  he  pays  to  Mr.  Smith,  and  has  1         Q        p 

no  shillings  left.     He  then  pays  £1  out  of  the  £4,  and  has 

£3  0  s.  8  d.  left  of  the  £4  4  s.  and  2  d.  which  he  had  at  first.  3        0        8 

On  comparing  this  process  with  the  one  represented  on 

page  42,  the  pupil  will  see  in  what  they  are  alike  and  in  what  respect  they  differ. 


SLATE     EXERCISES 

Find  the  differences 

2.  £         s.         d.        far.         3.     Bu.         pk, 

173     8       5       0  324      2 

75     9       7       3  235       3 


5.   Ed. 

yd. 

ft. 

in. 

15 

3 

2 

3 

8 

4 

1 

9 

6 

4 

0 

6 

1 

6 

qt. 

3 

7 

Pt. 
0 
1 

4.  Mi.    yd.    ft. 

17   1375   2 
7   938   2 

□  rd. 

35 
13 

d  yd.   d  ft.    d  in. 

14    6    81 
25    7   108 

21       18        7      117 
2        36 


6       4      2       0  21       19         1           9 

Notes. — 1.  In  Ex.  5,  when  we  come  to  the  yards,  we  say,  4  from  8 1/2  yd.  leaves 
4l/2  yd. ;  set  down  4,  and  for  the  1/2  yd.  add 

7.    Cwt.        lb.           oz.  i  ft.  6  in.  to  the  feet  and  inches  respectively. 

15        33        11/4  2.  In  Ex.  6,  when  we  come  to  the  sq.  yd., 

8  98        14  V*  we  say,  26  from  44  x/4  sq.  yd.  leaves  18  1/4  ;  set 
— rj 1Q3,  down  18,  and  for  the  J/4  sq.  yd.  add  2  sq.  ft. 

b        o4        1/6/4  an(i  35  sq.  in.  to  the  remainder. 

8.      Tr.       wk.         d.          h.  min.         sec.                9.       Bu.         pk.       qt.           pt. 

14  0       2       20  31       52                169       2       1       1% 

9  1       6       23  56       58                  71       3       7       1% 

10.    Gal.        qt.       pt.          gi  11.      Mi.           rd.          yd.       ft.         in. 

15  3       1       3%  26       230      4      2       10 


7      3       13%  19      309       5       2       ll7/. 


MEASURES.  227 

Multiplication. 

Example. — l.  Seven  bins  of  equal  dimensions  are  full  of  wheat. 
On  careful  measurement  one  of  them  is  found  to  contain  12  bu. 
3  pk.  5  qt.     How  much  is  there  in  the  7  bins  ? 

Explanation. — Seven  times  5  qt.  =  35  qt.  =  4  pk.  3  qt.  Written  Work. 

3  qt.  being  written  under  quarts  in  the  multiplicand,  the  4  Bu.       pk.      qt. 

pk.  are  added  to  seven  times  3  pk.     Seven  times  3  pk.  =  21  \<%        3        5 
pk.    21  pk.  +  4  pk.  =  25  pk.  =  6  bu.  1  pk.    Seven  times  12  „ 

bu.  =  84bu.    84  bu.  +  6  bu.  =  90  bu.    Hence  seven  times  12  

bu.  3  pk.  5  qt.  =  90  bu.  1  pk.  3  qt.  90        1        3 

Note. — The  pupil  should  obtain  the  result  also  by  addition,  and  thus  the  rela- 
tion of  addition  and  multiplication  will  be  more  deeply  impressed  on  his  mind. 
(See  also  page  55.) 

Explanation.— 12  times  5/8  in.  =  60/8  =  1 4/8 

2.  Yd.      ft-  in.  or  <7i/2  in.     12  x  7  in  =84  in.     84  in.  +  1l/2 

5        2  7  %  in.  =  91  yg  in.  =  1  ft.  7  l  /2  in.     1 l  /2  inches  being 

12  written  under  the  inches  of  the  multiplicand,  the 

^        :j  ^J7  rest  of  the  process  is  similar  to  that  explained 

'2  above. 

3.  Multiply  7  gal.  3  qt.  1  pt.  3  gi.  by  156. 

If  the  multiplier  is  large,  it  is  sometimes 
convenient  to  multiply  by  its  factors,  as  in 
this  case  by  13  and  12.  An  advantage  of  this 
method  is  that  all  the  written  work  is  preserved 
as  a  part  of  the  process.  One  process  may  be 
used  to  test  the  other. 

Examples.— 4.   Multiply  3  lb.    8  oz.    18  pwt.   8  gr.    by  35. 

(Employ  factors  of  35.) 

5.  Multiply  27  gal.  3  qt.  1  pt.  3  gi.  by  36  ;  by  236. 

6.  Multiply  17  wk.  4  d.  13  h.  27  min.  36  sec.  by  9  ;  by  79. 

7.  Multiply  23  cu.  yd.  6  cu.  ft.  459  cu.  in.  by  8;  by  72. 

8.  Multiply  6  lb.  8  oz.  15  pwt.  198/13  gr.  by  42  ;  by  84. 

9.  Multiply  ft,  9,   I  11,  3  7,  3  2,  gr.  17,  by  17 ;  by  36. 

10.  Multiply  1  ton  13  cwt.  73  lb.  9  oz.  by  65  ;  by  47. 

11.  Multiply  17  h.  47  min.  39  sec.  by  25  ;  by  124. 


Gal.          qt 

7      3 

pt.        gi. 

1         3 
13 

103       2 

0         3 
12 

1243       0 
18  pwt.   8 

1         0 
gr.    by 

228  STANDARD  ARITHMETIC. 

Division. 

Example. — l.  If  £13  8  s.  7  d.  is  equally  distributed  among  12 
boys,  how  much  does  each  receive  ? 

w  ...      yj.   ,  Explanation. — This  example  is  taken  from 

'  an  old  English  work.     It  is  accompanied  with  a 

1<i)sj16        o  S.         7  d.  happy  illustration,  from  which  the  following  is 

1        2  47/12  extracted : 

"  Suppose  a  gentleman  leaves  with  a  school- 
master £13  8  s.  1  d.  in  13  one  pound  notes,  8  shillings,  and  7  penny  pieces,  which 
he  orders  to  be  equally  divided  among  12  good  boys.  In  compliance  with  this  kind 
direction,  the  master  calls  the  12  boys  up  to  his  desk,  and,  in  the  first  place,  gives  to 
each  boy  a  one  pound  note.  Having  thus  disposed  of  12  notes,  he  changes  the  one 
which  remains  for  20  shillings,  and  adding  these  to  the  8  s.  given  him  by  the  gentle- 
man, he  has  now  28  shillings.  Out  of  these  he  gives  2  shillings  to  each  boy,  and 
has  4  shillings  left.  These  he  changes  for  48  penny  pieces,  and  putting  them  to 
the  7  which  he  had  at  first,  has  now  55,"  etc. 

Suggestion. — Let  the  pupil  complete  the  explanation,  exchanging  the  1  d.  which 
will  be  left  (of  the  55  d.)  into  farthings,  and  distributing  the  farthings.  He  will 
find  at  the  end  that  there  are  4  farthings  left.  Since  there  were  no  smaller  coins 
the  distribution  could  practically  go  no  further,  although  the  exact  fraction  is  stated 
mathematically  in  the  solution. — The  author  referred  to  gives  the  remainder  to  a 
poor  woman  who  is  going  by  at  the  moment. 


SLATE     EXERCISES 
2.       Bu.  pk.  qt.  pt.  3.  Lb-  oz.         pwt.         gr. 

5)25  3  7  1  63)  15         8         9         12 

5  bu.      0  pk.      6  qt.      %  pt. 

If  the  divisor  is  a  compound  quantity  of  the  same  kind  as  the  dividend  (that  is, 
if  we  wish  to  see  how  many  times  one  quantity  is  contained  in  another  of  the  same 
kind,  or  what  part  one  is  of  the  other),  we  first  reduce  dividend  and  divisor  to  the 
lowest  denomination  in  either,  and  then  proceed  as  in  simple  division. 

4.  How  many  times  does  86  bu.  3  pk.  7  qt.  1  pt.  contain  14 
bu.  lpk.  7qt.  l5/6  pt.? 

FIRST   STEr — REDUCTION. 

86  bu.  3  pk.  7  qt.  1  pt.  14  bu.  1  pk.  7  qt.  1%  pt. 

347  pk.  57  pk. 

2783  qt.  463  qt. 

5567  pt.  (dividend)  927%  pt.  (divisor) 


MEASURES.  229 


SECOND   STEP — DIVISION. 


Explanation. — When  seemingly  prepared  for  the 

y^J\/6JOOD/  division  it  is  perceived  that  the  lowest  denomination 

KKoiy\ooAf\i)  1S  •&***  °f  a  Pmt-     Hence  divisor  and  dividend  are 

OObfjdd4:U/4  both  re(jUced  to  sixths  of  a  pint  before  the  division  is 

4  n  , .  attempted. 

Answer. — 6  times.  r 

Examples.— 5.  Divide  878  wk.  4  d.  15  h.  37  min.  36  sec.  by  9. 

6.  Divide  4285  cu.  yd.  6  cu.  ft.  1689  cu.  in.  by  23  ;  by  85. 

7.  Divide  3964  lb.  9  oz.  15  pwt.  18  gr.  by  12  ;  by  97. 

8.  Divide  5863  gal.  3  qt.  1  pt.  3  gi.  by  8 ;  by  75. 

Find 

9.  %  of  3  tons  17  cwt.;  %  of  72  tons  13  cwt.  50  lb. 

10.  %  of  17  a  ft.  72  a  in.  ;  y144  of  7  cu.  ft.  576  cu.  in. 

11.  %  of  91  bu.  4  qt.  1  pt.  ;   %■  of  37  gal.  2  qt. 

12.  Vn  of  975  lb.  13  oz.  avoir.  ;   %  of  8  lb.  7  oz. 

13.  .66%  of  4  cu,  ft.  14  cu.  in.  ;   .875  of  1  cu.  yd.  15  cu.  ft 
1  cu.  in. 

14.  .125  of  126  cwt.  7  lb.  11  oz.  ;    %  of  83  gal.  3  qt.  1  pt. 

15.  How  many  bars  of  gold,  each  weighing  5  oz.  13  pwt.  21  gr. 
can  be  made  of  a  bar  weighing  1064  oz.  14  pwt.  15  gr.  ? 

16.  How  many  pieces  of  cord,  each  5%  yd.  long,  can  be  cut 
off  a  length  of  100  yards,  and  what  length  will  remain  ? 

17.  How  many  jars,  each  containing  2  gal.  3  qt.  1  pt.  3  gi., 
can  be  filled  out  of  a  cask  containing  285  gal.  ? 

18.  How  many  portions  of  time,  each  equal  to  1  day  7  h.  45 
min.  56  sec,  are  contained  in  346  d.  18  h.  34  min.  32  sec? 

19.  A  silver  ingot  (of  coin  standard)  weighing  175  oz.  13  pwt. 
9  gr.  contains  silver  enough  for  how  many  silver  dollars  ?   (P.  207.) 

20.  From  the  same  irgot  how  many  silver  quarters  could  be 
made  ?     How  many  dimes  ? 

21.  How  many  packages,  each  weighing  §  6,  3  4,  can  be  made 
from  a  quantity  of  medicine  weighing  fi>  17.25  ? 


230  STANDARD  ARITHMETIC. 

Adding  and  Subtracting  Denominate  Fractions. 

l.  Add  %  gal.  and  %  qt.  2.  Subtract  %  h.  from  %  d. 

First  reduce  the  fractions  to  integers,  then  proceed  as  above. 

Operation.  Operation. 

%  gal.  =  2  qt.  1  pt.  2/9  d.  =  5  h.  20  min. 

»/i  qt.    =  1  pt.  %  h.  =  50  min. 

3  qt.  0  pt.  4  h.  30  min. 

Examples.— 3.  Add  %  wk.,  %  d.,  and  %  h. 

4.  Add  Y,  cwk,  %  lb.,  and  %  oz. 

5.  Add  2%  bu.,  %  pk.,  and  y3  qt. 

6.  Add  7/9  gal.  and  y10  qt. 

7.  Add  £%,  %  s.,  and  8/10  d. 

8.  Subtract  yl6  h.  from  6/7  d. 

9.  Subtract  23/4  sq.  rd.  from  V/A  acre. 

10.  Subtract  %  oz.  from  2/5  lb.  Troy. 

11.  Subtract  %  pwt.  from  %  oz. 

12.  Add  Yie  cwt.  103/5  lb.  and  72/5  oz. 

13.  Add  3/5  of  a  ton,  %  of  a  cwt.,  and  %  of  a  lb. 

14.  5  Y,  miles  -  5%  fur.  +  33  %  rods  =  ? 

15.  5/32  n  mile  -f  7/io  acre  +  0. 75  p  rod  =  ? 

16.  4/7  of  a  wk.  +  3/5  of  a  day  +  5/e  of  an  h.+  y4  of  a  min.=  ? 

17.  Subtract  4/7  lb.  avoirdupois  from  4/5  lb.  Troy. 

(Find  the  result  in  grains.) 

18.  From  217/36  lb.  Troy  take  19/96  oz.  Troy. 

19.  Take  47/64  cwt.  from  1355/m2  T. 

20.  From  11%  wk.  subtract  8%  d.;   86/7  h. 

21.  Find  the  sum  of  4/7  cwt.,  85/6  lb.,  and  3%0  oz. 

22.  Find  the  difference  between  37/n  miles  and  3529/33  rd. 

23.  Add  3/5  wk.,  %  d.,  5/7  h.,  and  2/3  min. 

24.  Add  4/5  of  a  pound  avoirdupois  and  3/7  of  a  pound  Troy. 


Yr. 

mo. 

d. 

1879 

5 

7 

1868 

6 

12 

MEASURES.  231 

To  find  Difference  of  Time  between  Dates. 

Example. — l.  How  many  years,  months,  and  days  from  June 

12,  1868  to  May  7,  1879  ? 

Solution. — May  7,  1879,  is  the  7th  day  of  the  5th 
month  of  1879,  and  June  12  is  the  12th  day  of  the  ^6th 
month  of  1868.  Hence,  by  subtracting  1868  years  6 
months  and  12  days  from  1879  years  5  months  and  7 
days,  we  find  the  time  elapsed  from  the  earlier  to  the  later 
10,      10,      25,  date.     We  consider  30  days  a  month,  irrespective  of  what 

calendar  months  may  intervene  between  the  two  dates. 

24-8.  This  method,  though  usually  employed  in  business, 
does  not  obtain  the  exact  time  elapsing  from  one  date  to  another. 
A  more  accurate  method  is  to  find  first  the  number  of  entire  years 
between  the  dates,  then  of  entire  months,  and  lastly  of  the  days. 

The  difference  between  the  results  of  these  methods  may  be  seen  from  solutions 
of  the  following  problem  : 

2.  A  sum  of  money  was  borrowed  Sept.  18,  1867,  and  paid 
March  16,  1870.     How  long  was  it  in  the  hands  of  the  borrower  ? 

First  Method. 
1870  yr.     3  mo.     16  d. 
1867  "      9  "        18  " 
2  yr.     5  mo.     28d7 
Second.  Method. 
From  Sept.  18,  1867,  to  Sept.  18,  1869  =  2  yr. 
"    Sept.  18,  1869,  to  Feb.  18,  1870  =  5  mo. 
"    Feb.  18  to  March  16  =  26  d. 

Total :  2  yr.  5  mo.  26  d. 

249.  The  first  method  is  based  on  the  supposition  that  there 
are  360  days  in  the  year,  and  30  days  in  each  month. 

The  second  method  takes  no  account  of  the  number  of  days  in 
the  several  years  nor  in  the  entire  month,  but  reckons  a  year 
from  a  given  day  of  one  year  to  the  corresponding  day  of  the  next, 
and  a  month  from  a  given  day  of  one  month  to  the  same  day  of 
the  next.  In  reckoning  the  odd  days,  however,  it  takes  into  ac- 
count the  number  of  days  in  the  month  preceding  the  last 


232  STANDARD  ARITHMETIC. 

250,  To  find  the  exact  number  of  days  between  two  dates,  we 
must  add  together  the  number  of  days  in  the  several  years,  allow- 
ing 366  days  for  a  leap  year,  then  the  number  of  days  in  the  odd 
months,  according  to  the  calendar,  and  finally  the  number  of  re- 
maining days. 

Third  Method. 

m  ,  j  From  Sept.  18,  1867,  to  Sept.  18,  1868  =  366  days, 

wnoie  years. -j      „     Sept.  18, 1868,  to  Sept.  18,  1869  =  365     " 
Remaining     j  Sept.  12,  Oct.  31,  Nov.  30,  Dec.  31, 

days.  (     Jan.  31,  Feb.  28,  Mar.  16.  =  179     " 

The  exact  time  in  days  =  910  days. 

Examples.— Find  the  interval  of  time  between  the  following 
dates  by  the  first  method  : 

1.  Feb.  3,  1845,  and  Dec.  17,  1852. 

2.  Oct.  19,  1871,  and  Nov.  1,  1873. 

3.  Apr.  2,  1876,  and  Jan.  31,  1881. 

4.  Sept.  30,  1872,  and  July  2,  1879. 

5.  How  many  years,  months,  and  days  from  the  Declaration 
of  Independence  to  the  surrender  of  Cornwallis,  at  Yorktown, 
1781,  Oct.  19  ? 

6.  Find  the  preseut  age  of  the  American  Republic,  born  4th 
of  July,  1776. 

7.  Washington  was  born  Feb.  22,  1732,  and  lived  67  yr.  9  mo. 
22  d.     At  what  date  did  he  die  ? 

8.  Find  the  exact  number  of  days  of  your  own  life. 

9.  A  person  born  Dec.  8,  1845,  died,  aged  36  years,  1  mo. 
18  d.     What  was  the  date  of  his  death  ? 

10.  General  Grant  died  July  23,  1885,  at  the  age  of  63  yrs.  2 
mo.  26  d.     What  was  the  date  of  his  birth  ? 

11.  Abraham  Lincoln  died  Apr.  15,  1865,  and  General  Gar- 
field Sept.  19,  1881.  By  what  length  of  time  did  the  death  of 
each  precede  that  of  General  Grant  ? 


MEASURES.  233 

Longitude  and  Time. 

251.  The  line  which  may  be  supposed  to  be  drawn  from  pole 
to  pole  through  any  place  on  the  surface  of  the  earth  is  called  the 
meridian  (mid-day  line)  of  that  place.  All  places  having  the 
same  meridian  have  their  noon  at  exactly  the  same  moment. 

Since  the  earth  revolves  upon  its  axis  from  west  to  east,  the 
sun  seems  to  come  from  the  east  to  each  meridian  successively, 
and  thus  to  go  around  the  earth  from  east  to  west  in  24  hours. 

The  circumference  of  the  earth  is  divided  into  360  degrees 
(360°),  and  inasmuch  as  it  revolves  once  in  24  hours,  15°  must 
pass  under  the  sun  in  one  hour  ;  y60  of  15°  =  74°  =  15'  of  circum- 
ference in  1  minute  of  time,  and  y6o  of  15'  =  *//  =  15ff  of  circum- 
ference in  1  second  of  time.  Hence  we  have  the  following  table, 
showing  the  correspondence  of  longitude  and  time. 

Table  of  Longitude  and  Time. 

360°  of  Longitude  correspond  to  24  hours  in  time. 
15°  of         "  "  1  hour  in  time. 

15'  of         "  "  1  min.  in  time. 

15"  of         "  "  1  sec.  in  time. 

Note.— If  three  clocks,  all  keeping  correct  time,  be  placed  at  the  distance  of 
15°  longitude  from  each  other,  the  one  farthest  east  will,  at  any  moment,  be  found 
one  hour  faster,  and  the  one  farthest  west  one  hour  slower,  than  the  clock  midway 
between  them. 

ORAL    EXERCISES. 

1.  If,  in  traveling,  I  find  my  watch,  which  is  a  reliable  time- 
keeper, growing  faster  and  faster  as  compared  with  the  time  in 
the  places  through  which  I  pass,  should  I  conclude  that  I  am 
traveling  eastward  or  westward  ? 

2.  If  I  find  my  watch  an  hour  fast,  how  many  degrees  have 
I  gone,  and  in  which  direction  ?  If  I  find  it  half  an  hour  fast  ? 
15  minutes  ? 

3.  How  many  degrees  of  the  earth's  surface  pass  under  the 
sun's  vertical  rays  in  2,  4,  7,  13,  19,  21  hours  ? 


234  STANDARD  ARITHMETIC. 

4.  I  start  from  Cincinnati.,  and,  on  arriving  at  another  city, 
compare  my  watch  with  a  well-regulated  clock,  and  find  it  faster 
than  my  watch  ;  have  I  traveled  east  or  west  ?  I  find  it  20  min- 
utes faster ;  how  many  degrees  have  I  traveled  ? 

5.  When  it  is  12  o'clock  noon  at  Omaha,  what  is  the  time 
at  places  lying  15,  7 1/2f  3  %  degrees  west  ?    East  ? 

6.  What  difference  in  longitude  makes  a  difference  of  1  hour 
30  min.  in  time?    Of  1%  min.?     Of  V/2  sec? 

7.  Suppose  the  sun  is  rising  at  4  o'clock  A.  m.  on  the  first  me- 
ridian (Greenwich),  on  what  meridian  is  it  noon  ? 

8.  What  is  the  difference  of  time  between  Greenwich  (on  the 
first  meridian)  and  a  place  lying  under  the  74th  meridian  ? 


SLATE     EXERCISES 


9.  What  is  the  difference  of  longitude  between  Washington  and 
Cleveland,  the  difference  in  time  being  18  min.  32  sec.  ? 

Explanation. — One  second  in  time  corresponds  to 
a  difference  of  15"  of  longitude,  hence  32  sec.  in  time 
18  mm.  32  sec.  correspond  to  32  times  15"  in  long.  cs  480"  =  8'  in  long. 

15  One  min.  in  time  corresponds  to  15'  in  long.,  hence 

to  oqi       '     a  18  min.  in  time  correspond  to  18  times  15'  in  long.  = 

270'  of  longitude.     270'  +  8'  =  278'  ==  4°  38'  of  longi- 
tude. 

Note. — Let  it  be  kept  in  mind  that,  inasmuch  as  there  are  360°  in  the  circum- 
ference of  the  earth,  and  only  24  h.  in  the  day,  there  are  more  degrees  in  any  differ, 
ence  of  longitude  than  hours  in  difference  of  time ;  more  minutes  of  longitude  than 
minutes  in  time,  and  more  seconds  of  longitude  than  seconds  in  time.  (How  many 
times  as  many  ?) 

10.  When  it  is  noon  at  Washington,  it  is  only  11  o'clock,  17 
min.  and  44  sec.  A.  m.  at  Chicago.  Find  (a)  the  difference  in 
time  ;  (I?)  the  difference  in  longitude. 

To  find  Difference  in  Time.  To  find  Difference  of  Longitude. 

a.  12  h.        0  min.    0  sec.  h.        42  min.  16  sec. 

11  17  44  15 


42  min.  16  sec.  10°     34'  0" 


MEASURES. 


235 


252.  The  names  of  a  few  important  cities  are  given  below,  with 
the  longitude  of  each  from  Greenwich  (see  Hasweli,  ed.  1885): 


Cities. 

Longitude. 

Cities. 

] 

Jongitude. . 

Albany,  N.  Y. 

73° 

45   24" 

W. 

New  Orleans,  La. 

90° 

3'  28"  W. 

Berlin,  Germ. 

13° 

23'  45" 

E. 

New  York,  N.  Y. 

74° 

0  24"  W. 

Boston,  Mass. 

71° 

3'  30" 

W. 

Paris,  France. 

2° 

20     0"  E. 

Calcutta,  India.  88° 

20     0' 

E. 

Philadelphia,  Pa. 

75° 

9     3"  W. 

Chicago,  111. 

87° 

37'  47' 

W. 

Rome,  Italy. 

12° 

27'    0"  E. 

Cincinnati,  0. 

84° 

29'  45" 

W. 

St.  Louis,  Mo. 

90° 

12'     4"  W. 

Cleveland,  0. 

81° 

40'  30" 

W. 

St.  Petersburg",  Russ.  30 

°  19'  0"  E. 

London,  Eng. 

0° 

0     0" 

San  Francisco,  Cal. 

122° 

23  19"  W. 

(Greenwich.) 

Washington,  D.  C. 

77° 

0   3«"  W. 

11.  When  it  is  noon  at  St.  Louis,  is  it  before  or  after  noon  at 
Washington  ?    At  San  Francisco  ? 

12.  When  it  is  noon  at  San  Francisco,  is  it  before  or  after 
noon  at  St.  Louis  ?    At  Washington  ? 

13.  When  noon  in  St.  Louis,  about  what  time  in  Calcutta  ? 

14.  When  it  is  midnight  at  Paris,  what  is  the  local  time  at 
Cleveland  ? 


Solution. 
{1st  step.) 
Cleveland      81°  40'  30" 
Paris  2    20 


84°     0' 

30" 

15)84< 

{2d  step.) 
'     0'    30' 

i 

5  1 

i.  36  min. 

2  sec. 

12  h. 

5 

{3d  step.) 
0  rain. 
36 

0  sec. 
2 

6  23  58 

Ans.— 23  min.  58  sec.  past 
6,  r.  m. 


Explanation. — First  step. — To  find  the  dif- 
ference of  longitude  we  add  the  longitude  of 
Paris  to  that  of  Cleveland,  since  Paris  lies  east 
and  Cleveland  west  of  Greenwich. 

Second  step.— Since  15°  of  longitude  pro- 
duce a  difference  of  1  hour  in  time,  84°  pro- 
duce a  difference  of  5  hours  in  time  and  some- 
thing more,  for  there  are  9°  more  than  5  times 
15°  in  84°. 

The  remainder,  9°  =  9  x  60'  =  540',  which 
being  added  to  the  2'  in  the  next  term,  we 
have  542',  etc.  (The  pupil  should  be  able  to 
take  the  remaining  steps  of  the  analysis  with- 
out aid.) 

Third  step. — Since  Cleveland  is  west  from 
Paris,  the  time  of  Cleveland  is  earlier  than 
that  of  Paris,  henco  we  subtract  5  h.  36  m. 
2  sec  from  12  h. 


236  STANDARD  ARITHMETIC. 

Rules  for  Computations  in  Longitude  and  Time. 

I.  For  finding  difference  of  longitude,  difference  in  time  be- 
ing given  : 

253.  Rule, — Multiply  the  difference  in  time  by  15,  and  the 
hours,  minutes,  and  seconds  of  time  will  give  respectively  °s,  's, 
and  "s  of  longitude. 

II.  For  finding  difference  in  time,  difference  of  longitude  be- 
ing given  : 

254.  Rule, — Divide  the  difference  of  longitude  by  15,  and  the 
°s,  's,  "s  of  longitude  will  give  respectively  hours,  minutes,  and 
seconds  of  time. 


SLATE     EXERCISES. 

Examples. — Using  the  longitudes  given  in  the  above  table, 

Find  the  difference  in  time  between 


1.  Albany  and  Chicago.     6.  St.   Louis   and  San 

10.  New  York  and  New 

2,  Berlin  and  Paris.                   Francisco. 

Orleans. 

3.  Greenwich   and  St.      7.  Home  and  Paris. 

11.  Washington        and 

Petersburg.               8.  Washington      and 

Philadelphia. 

4.  Boston    and   Cleve-              Calcutta. 

12.  San    Francisco     and 

land.                           9.  Cincinnati   and  San 

Calcutta. 

5.  Boston  and  St.  Louis.             Francisco. 

For  oral  exercises,  let  the  differences  in  time  be  estimated.     Rough  estimates 

are  as  frequently  required  as  exact  computations. 

13.  When  it  is  12  o'clock  noon  at  Greenwich,  what  is  the  time 
at  each  of  the  cities  mentioned  in  the  table  (Art.  252)  ? 

14.  Suppose  it  to  be  8  o'clock  p.  m.  (Post  meridian  or  after 
noon)  at  Berlin,  what  time  is  it  at  the  other  cities  ? 

15.  Suppose  it  to  be  7  o'clock  A.  m.  (Ante  meridian  or  before 
noon)  at  New  York,  what  time  is  it  at  the  other  cities  ? 

16.  The  difference  in  time  between  two  places  is  1  b.  15  min. 
2G  sec. ;  what  is  their  difference  of  longitude  ? 

17.  Find  the  distance  in  geographical  miles  between  two  places 
on  the  equator  that  are  3  h.  2  min.  12  sec.  apart.   (1°  =  60  geo.  mi.) 


MEASURES.  237 

Applications  and  Review. 

1.  If  1  cwt.  costs  $16.16,  $18.25,  $19.50,  $25.25,  $36.36,  what 
is  the  cost  of  328  lb.  ? 

2.  If  1  gallon  costs  $24/5,  $3.50,  $1%,  $3.30,  $4%,  $2%,  what 
is  the  cost  of  5  y4  gallons  ? 

3.  If  1  pound  costs  7y2  francs,  what  is  the  cost  of  3%,  2.5, 
1%,  4.6,  5%  pounds? 

4.  If  1  cwt.  costs  $127.64,  what  is  the  cost  of  %%  6%,  9%, 
17y8  lb.  avoirdupois? 

5.  What  will  a  lot,  measuring  57y2Xl00  feet,  cost,  if  the 
price  of  1  n  foot  is  $25%,  $37%,  $48  %  $57%? 

6.  What  will  be  the  cost  of  5%,  8%,  15 7*  23%  oz.,  if  1  oz. 
cost  37%**? 

7.  What  does  a  family  spend  for  meat  in  a  month  of  30  days, 
at  433/4^  a  day  ?    In  a  year  of  365  days  ? 


8.  If  5  gal.  cost  $1.15,   $2.35,   $4.25,   $6.75,   $7.45,   $15.25, 
$20.20,  what  is  the  cost  of  1/2  gallon  ?    (One  step.) 

9.  If  4%  lb.  cost  $2.32,   $3.48,   $4.28,   $5.44,   $6.64,  $7.22, 
$9.04,  what  is  the  cost  of  %>  lb.? 

10.  If  3%  dozen  cost  $14.48,  $28.36,  $30,  $16.75,  $27.50,  what 
is  the  cost  of  1  gross  ? 

11.  If  6  reams  of  writing-paper  cost  $7.20,    $18.50,    $20.25, 
$30.75,  $50.15,  what  is  the  cost  of  18  quires  ? 

12.  If  9  acres  cost  $100.75,  $140.25,  $225.50,  $350.40,  what 
is  the  cost  of  67.5  acres  ? 

13.  If  7  barrels  cost  $48.25,  $36.70,  $64.83,  $94.24,  what  is  the 
cost  of  y2  barrel  ? 

14.  If  8  cords  cost  $24.25,  $25.75,  $26.40,  $38.85,  what  is  the 
cost  of  y4  cord  ? 

15.  If  1  cwt.  costs  $568.25,  what  will  20  lb.  cost?  25  lb.  ? 

33y3  lb.  ?     (Pursue  the  shortest  method.) 
11 


238  STANDARD  ARITHMETIC. 

1.  If  %  gal.  costs  520,  what  will  2,  5,  7,  17,  46  gal.  cost  ? 

Analysis.— If  1/6  gal.  costs  520,  1  gal.  will  cost  5  x  520  =$2.60,  and  if  1 
gal.  costs  $2.60,  2  gal.  will  cost  2  x  $2.60  =  $5.20. 

2.  If  Vb  bu.  costs  800,  what  will  5,  9,  13,  23  bu.  cost  ? 

3.  If  ys  dol.  is  paid  per  hour  for  labor,  what  is  paid  for  19, 
23  h.?    For  2%  days,  10  h.  per  day  ?    For  3  days  ? 

4.  If  yi0  gal.  costs  480,  what  will  7,  18,  32,  21  gallons  cost  ? 

5.  If  yi3  lb.  Troy  costs  $%,  what  will  9,  11,  19,  35  oz.  cost  ? 

6.  If  8  oz.  avoirdupois  cost  $28.30,  what  will  17,  73,  85,  99 
lb.  cost  ? 

7.  If  %  quire  costs  %  fr.,  how  many  francs  will  8,  13,  23,  45 
quires  cost  ? 

8.  If  %  lb.  costs  $.07,  $.09,  $.11,  what  is  the  cost  of  6  lb.  ? 

9.  If  %  doz.  pens  cost  y10  dol.,  what  will  6,  9,  17,  28  doz. 
cost  ? 

10.  If  1  qt.  costs  10&  120,  180,  what  will  18  bu.  cost  ? 

11.  If  %  doz.  costs  $.60,  $.40,  $.30,  $.20,  what  will  1  gross 
cost? 

12.  If  %  doz.  costs  $y6,  $y3,  $y8,  $y9,  what  will  1  gross  cost  ? 


13.  If  l3/4  thousand  shingles  cost  $62/3,  what  will  28  M  cost  ? 

14.  If  6  oz.  Troy  cost  9/10  dol.,  what  will  5,  8,  30  lb.  cost  ? 

15.  If  1%  bu.  cost  $27/8,  what  will  19,  31,  84,  73  bu.  cost  ? 

16.  If  4%  bl.  cost  $105,  what  will  8,  7%'  ll3/4  bl.  cost? 

17.  If  3%  cords  cost  $24 %  what  will  9,  12,  19  cords  cost  ? 

18.  If  3%  oz.  cost  $.70,  what  is  the  cost  of  5,  8,  17  oz.? 

19.  If  1%  pk.  cost  $1%,  what  is  the  cost  of  iy4,  2%,  33/4  bu.  ? 

20.  If  3%  cwt.  cost  $46%,  what  is  the  cost  of  33/4,  61/?  cwt.? 

21.  If  1%  doz.  cost  $l3/4,  what  is  the  cost  of  2%,  7%  doz.? 

22.  If  4.5  yd.  cost  $12.30,  what  is  the  cost  of  7.7,  9.13  yd.? 

23.  If  3.48  lb.  cost  $1.24,  what  is  the  cost  of  2.36,  9.81  lb.? 


MEASURES.  239 

Square  and  Cubic  Measures. 

Examples. — l.  How  many  square  yards  of  oil-cloth  will  cover 
a  floor  14  ft.  long  and  12  feet  wide  ? 

Analysis.— The  area  of  the  floor  is  14  x  12  ft.  =  168  □  ft.,  =  182/3  □  yd., 
hence  182/3   □  yd.  of  oil-cloth  will  be  required  to  cover  the  floor. 

2.  How  many  acres  in  a  roadway  100  rods  long  and  18  yards 
wide  ? 

3.  How  many  bricks  will  it  take  to  pave  a  sidewalk  32  ft.  long 
and  6  ft.  wide,  there  being  4%  bricks  to  the  □  foot  ? 

4.  How  many  yards  of  paper  will  be  needed  to  paper  a  room 
14  ft.  long,  12  ft.  wide,  and  9  ft.  high,  if  the  paper  is  18  in. 
wide,  no  deduction  being  made  for  doors  and  windows  ? 

5.  How  many  rolls  of  paper  of  8  yd.  each  will  be  needed  to 
paper  a  room  18  ft.  long,  15  ft.  wide,  and  10  ft.  high,  if  the  paper 
is  18  in.  wide,  and  one  roll  is  saved  by  the  windows  and  doors  ? 

Explanation. — Rolls  of  wall-paper  24  ft.  long  would  make  2  strips  each  10  ft. 
long,  the  strips  not  being  pieced ;  but,  if  the  4  ft.  left  were  used  under  and  over  the 
openings,  we  would  need  to  know  how  many  times  the  surface  of  a  roll  is  contained 
in  the  surface  of  the  walls.     (For  paper-hangers'  method,  see  p.  266.) 

6.  How  many  □  yd.  of  plastering  are  required  for  a  room  20 
ft.  long,  15  ft.  wide,  and  10  ft.  6  inches  high  ? 

7.  How  many  shingles  will  it  take  to  cover  both  sides  of  a  roof, 
the  rafters  of  which  are  16  ft.  long  and  the  ridge-pole  is  23  ft. 
long,  if  each  shingle  has  an  area  of  162  □  inches,  but  %  of  it  is 
covered  by  other  shingles  ? 

8.  How  many  cords  of  wood  in  a  pile  36  ft.  long,  10  ft.  6  in. 
high  and  4  ft.  wide  ? 

Analysis. — Length  x  width  x  height  =  no.  cubic  ft.,  and  128  cubic  feet  are 
equal  to  1  cord. 

9.  How  many  cords  of  wood  in  a  pile  50  ft.  long,  11  ft.  3  in. 
high,  and  5  ft.  wide  ? 

10.  How  many  cubic  feet  in  a  room  16  ft.  long,  14  ft.  wide, 
and  9  feet  high  ? 


66  ft. 


240  STANDARD  ARITHMETIC. 

11.  What  is  the  capacity  in  gallons  of  a  vat  10  ft.  long,  3  ft. 
wide,  and  4  feet  deep  ? 

12.  How  many  cu.  ft.  in  a  square  tank,  2y3  yd.  wide  and 

long,  and  8  ft.  6  in.  deep  ? 

13.  How  many  qt.  of  milk  can 
be  put  into  a  can  containing  1496 1/2 
cu.  in.? 

Remember,  a  gallon  fills  the  space  of  231 
cubic  inches. 

14.  How  many  loads  of  earth  must 
4i  ft.  8  in.      '         be  removed  in  digging  a  cellar  to  the 

depth  of  6  ft.,  and  of  other  dimen- 
sions as  given  in  the  diagram  ?     (A  load  is  estimated  to  be  1  cubic  yard.) 

15.  How  much  will  it  cost  to  pave  the  floor  of  this  cellar  at 
14^  per  □  foot  ?  How  may  bricks  will  it  require  if  laid  on  edge 
7  to  the  □  foot  ? 

16.  How  many  square  inches  in  the  largest  circle  that  can  be 
cut  from  a  card-board  2  ft.  square  ?    (See  note,  page  243.) 

17.  A  tank  is  5  ft.  6  in.  long,  5  ft.  3  in.  wide,  and  6  ft.  8  in. 
deep.     How  many  gallons  will  it  hold  ?    How  heavy  is  the  water 

Contained  in  it,  if  3  ft.  deep  ?     (A  cubic  foot  of  pure  water  at  62°  weighs 
997.68  oz.) 

18.  If  a  horse  can  draw  1600  lb.  on  a  given  road,  how  many 
cubic  feet  of  lead  can  2  horses  draw  on  the  same  road,  a  cubic 
foot  of  lead  weighing  709.5  lb.?  How  many  men  whose  average 
weight  is  165  lb.  12  oz.  ? 

19.  A  cubic  foot  of  ice  at  the  temperature  of  32°  weighs  57.5 
lb.  How  many  tons  can  be  stored  in  an  ice-house  that  is  80  ft. 
long,  30  ft.  9  in.  wide,  and  20  ft.  deep  ? 

20.  How  many  paper  boxes  3  in.  long,  2  in.  wide,  and  2  in. 
deep,  can  be  packed  in  a  box  3  ft.  long,  2  ft.  wide,  and  V/g  ft. 
deep  ? 

21.  How  many  cubic  inches  in  a  can  holding  16  gal.  1  pt.? 


MEASURES.  241 

22.  How  many  square  feet  in  the  floor  of  your  school-room  to 
each  pupil  present  ? 

23.  At  3y8#  a  d  yd.,  how  much  will  it  cost  to  have  3  ceil- 
ings kalsomined,  each  measuring  15  by  14 1/2  ft.? 

24.  At  the  same  rate,  what  will  it  cost  for  kalsomining  the 
ceiling  and  walls  of  a  room  16  ft.  long,  15  ft.  wide,  and  10  %  ft. 
high,  allowing  8y8  □  yd.  for  doors  and  windows  ? 

Analysis. — The  ceiling  contains  16  x  15  ft.  =  240  □  ft. 

Two  walls  contain  each  16  x  lO1^  ft.  =  336  □  ft. 

Two  walls  contain  each  15  x  lO1/^  ft.  =  315  □  ft. 

99  □  yd.  -  8l/2  □  yd.  =  901/*  a  yd.  891   □  ft.  or  99  □  yd. 

At  S1/^  what  will  90 Va   0  yd-  cost? 

25.  What  will  it  cost  to  paint  a  room  24  ft.  long,  20  */,  ft. 
wide,  and  12  ft.  high,  at  10 1/2<f  a  a  yd.,  taking  out  4  □  yd. 
for  windows  ? 

26.  What  will  it  cost  to  paper  a  room  of  the  same  dimensions 
as  those  given  in  Example  24,  if  the  paper  is  18  in.  wide,  and  a 
roll  of  it,  measuring  8  yd.  in  length,  costs  22^.  The  border 
costs  3<fi  a,  yard. 

27.  What  will  the  laying  of  a  flag- stone  walk,  85  ft.  long  and 
5  ft.  wide,  cost  at  $2. 15  a  n  yd.  ? 

28.  What  will  it  cost  to  have  the  roof  of  a  house  shingled, 
the  rafters  of  which  are  16  ft.  long  and  the  ridge-pole  25  ft.  long, 
if  the  □  yard  costs  $.50  ? 

29.  What  will  it  cost  to  have  a  tin  roof  put  on  my  stable,  each 
slope  of  which  measures  20  by  14  ft.,  at  $5.75  per  100  n  ft.? 

30.  How  many  bricks  are  required  to  pave  a  cellar  36  ft.  long 
and  24y2  ft.  wide,  a  brick  measuring  8  by  4  inches  ?  What  will 
be  the  cost  of  the  job  if  I  am  charged  $1.65  per  o  yard  ? 

31.  What  will  it  cost  to  fill  a  jug,  which  contains  2310  cubic 
inches,  with  vinegar,  at  1$  per  quart  ? 

32.  The  same  telegram  is  sent  direct  from  New  York  City,  at 
12  o'clock  noon,  to  Cleveland,  Cincinnati,  St.  Louis,  and  San 
Francisco.     At  what  time  should  it  be  received  in  each  city  ? 


242  STANDARD  ARITHMETIC. 

Miscellaneous  Problems. 

1.  Fred  paid  $8. 50  per  week  for  board  from  April  3,  1883,  to 
May  8,  1884  ;  how  much  did  his  board  cost  him  for  that  period  ? 

2.  Three  lb.  of  sugar  are  needed  for  canning  5  qt.  strawberries  ; 
how  many  lb.  of  sugar  are  required  for  3  %  bu.  of  berries  ? 

3.  Five  francs  are  equal  to  4  marks ;  what  are  500  francs 
worth  in  German  money  ?  (Estimate.)  What  are  600  marks  worth 
in  French  money  ?     (Estimate.) 

4.  Of  104.688  lb.,  26.9  lb.  were  sold  yesterday,  and  %  of  the 
remainder  to-day.     How  many  lb.  and  oz.  are  left  ? 

5.  What  is  the  general  estimate  for  1,000,000  francs,  in  U.  S. 
money  ?  For  £240,000  ?  What  are  the  exact  values  of  these 
sums  ? 

6.  A  wall  1690  feet  long  is  to  be  built  in  30  days,  and  it  is 
found  that  7  men  in  14  days  have  completed  only  490  feet ;  how 
many  additional  men  must  be  employed  that  the  wall  may  be 
completed  in  the  required  time  ? 

7.  A  grocer  bought  goods  done  up  in  pound  packages.  On 
weighing  them  he  found  each  to  be  one  ounce  short.  How  much 
should  be  deducted  if  $45  was  the  charge  for  the  whole  ? 

8.  If  nine  geese  will  yield  4  lb.  8  oz.  of  bed-feathers,  how 
many  lb.  will  24  geese  yield,  at  the  same  rate  in  the  same  time  ? 

9.  The  ocean  covers  .734,  the  land  .266,  of  the  surface  of  the 
earth.  How  many  times  the  surface  of  the  land  is  the  surface 
of  the  water  ? 

10.  In  the  northern  hemisphere  of  the  earth,  the  land  covers 
.4  of  the  surface,  while  in  the  southern  hemisphere  it  covers 
.12:  How  many  times  as  much  land  in  the  northern  hemisphere 
as  in  the  southern  ? 

11.  Dividing  the  surface  of  the  earth  into  1000  equal  parts, 
398.7491  of  these  parts  are  in  the  torrid  zone,  259.1555  in  each 
of  the  temperate  zones.  How  many  are  in  the  arctic  and  antarc- 
tic zones  ? 


MEASURES.  243 

12.  Sound  travels  1090  ft.  per  sec.     How  far  in  .16%  min.  ? 

13.  Three  persons  together  buy  a  quantity  of  butter,  weighing 
260  lb.,  for  $92%.  The  first  takes  1  cwt.,  the  second  90  lb.,  and 
the  third  the  remainder.     How  much  does  each  have  to  pay  ? 

14.  Of  a  certain  kind  of  cloth,  29  in.  wide,  12  yd.  are  required 
for  a  dress.  How  many  yd.  would  be  required  if  the  cloth  were 
35  inches  wide,  provided  the  two  kinds  cut  to  equal  advantage  ? 

15.  A  lady  bought  a  quantity  of  butter  weighing  24  lb.  How 
many  ounces  are  used  per  day,  if  the  whole  quantity  lasts  her 
family  from  the  first  of  May  to  the  15th  June  ? 

16.  A  cwt.  of  salt  cost  $4 ;  what  is  the  cost  of  1  lb.  ?  Of  4  oz.  ? 
Of  500  lb.  ?     Of  a  ton  ?     Of  200  lb.  ? 

17.  If  a  thousand  cigars  cost  $85  ;  what  is  the  cost  of  3  boxes, 
each  containing  50  ?  How  much  does  a  man  pay  for  cigars  who 
smokes  3  per  day  for  one  year  ? 

18.  How  many  lb.  of  butter  can  be  made  in  1  week  from  the 
milk  of  12  cows,  giving  an  average  of  12  qt.  1  pt.  each  daily,  if 
25  qt.  yield  1  lb.  8  oz.  butter  ? 

19.  How  much  per  hour  do  I  pay  the  laborer  who  works  3% 
days,  8  hours  a  day,  and  receives  $5  for  the  time  ? 

20.  A  family  agreed  to  pay  rent  at  the  rate  of  $190  a  year,  but 
after  7  %  months  left  the  premises,  with  consent  of  the  proprietor. 
How  much  should  they  pay  ? 

21.  William,  walking  briskly,  goes  6600  paces  per  hour.  He 
reaches  the  next  village  in  40  minutes.  How  many  paces  distant 
is  the  place  ?    How  many  miles  if  each  pace  is  2.7  ft.  ? 

22.  If  the  distance  from  one  place  to  another  is  16  miles  80 
rods,  how  far  is  it  to  an  intermediate  place,  the  latter  distance  be- 
ing %  of  the  former  ? 

23.  Find  the  area  of  a  circle  in  □  yd.,  if  its  diameter  is  4.05 
yards. 

Note. — The  area  of  a  circle  is  .7854  of  the  area  of  a  square  whose  side  is 
equal  to  the  diameter  of  the  circle. 


244  STANDARD  ARITHMETIC. 

24.  A  man  saw  the  flash  of  a  cannon,  which  was  discharged 
in  the  distance,  5%  seconds  before  he  heard  the  report.  How 
far  was  he  from  the  cannon  ?    (See  Ex.  12.) 

25.  Between  the  lightning  and  the  thunder  I  noted  9% 
seconds  ;  how  far  away  was  the  thunder-cloud  ? 

26.  Ernest  bought  3  yd.  of  broadcloth  for  $11,  and  when  he 
took  it  to  the  tailor  to  have  a  coat  made  of  it,  he  found  that  he 
had  to  get  %  yd.  more.     How  much  did  the  cloth  cost  him  ? 

27.  If  the  knitting  of  a  pair  of  stockings  costs  24^,  and  a  lb. 
of  worsted  costs  $1.10,  what  will  a  pair  of  stockings  cost  for  which 
4  ounces  of  worsted  are  used  ? 

28.  A  family  uses  daily  4/5  lb.  of  butter  at  2^  an  oz.,  and  iy2 
qt.  of  milk  at  4^  a  pint.  How  much  do  the  butter  and  milk  cost 
the  family  per  month  of  30  days  ? 

29.  A  farmer  sold  25  bu.  3  pk.  of  pears  at  45  ^  per  bu.  How 
many  yards  of  calico  at  8y2^  can  he  obtain  for  the  proceeds  ? 

30.  In  making  strawberry- jam,  1  lb.  2  oz.  of  sugar  are  com- 
monly taken  to  1  lb.  4  oz.  of  fruit.  Find  the  amount  of  sugar 
required  for  38  lb.  of  berries. 

31.  In  making  jelly  we  commonly  take  1  lb.  of  sugar  to  1  pint 
of  juice.     Find  the  amount  of  sugar  required  for  17  Vs  "jt  juice. 

32.  A  grocer  bought  a  barrel  of  vinegar  containing  123  qt., 
at  120  a  gallon,  and  paid  for  it  with  coffee  at  37%^  a  lb.  How 
many  lb.  did  it  require  ? 

33.  What  will  it  cost  to  have  a  manuscript  of  8  quires  16  sheets 
copied,  at  15^  a  page,  one  side  only  of  each  leaf  to  be  written  on  ? 

34.  How  many  cubic  feet  of  masonry  in  a  wall  33  yd.  long, 
10  ft.  high,  and  1  ft.  6  in.  thick  ? 

35.  How  many  cubic  inches  in  a  tank  9  ft.  6  in.  long,  5  ft. 
3  in.  wide,  and  3y3  ft.  deep  ?    How  many  gallons  ? 

36.  Awheel  turns  300.5  times  in  6  min.  15  sec;  how  many 
times  in  1  min.  ? 


MEASURES.  245 

37.  Our  earth  completes  its  circuit  around  the  sun  in 
365.242199  days,  and  thereby  travels  a  distance  of  129847287.467 
geogr.  miles.     What  distance  does  it  travel  in  1  day  ?    In  1  sec.  ? 

38.  A  commission  merchant  wishes  to  ship  1384  bu.  of  grain 
in  sacks,  each  holding  2  bu.  3  pk. ;  how  many  sacks  does  he  need  ? 

(1  sack  for  any  remainder.) 

39.  A  railroad  track  has  a  grade  of  35  ft.  9  in.  to  the  mile. 
In  how  many  miles  will  the  rise  amount  to  143  ft.  ? 

40.  Mr.  M.  received  3  shipments  of  goods,  namely,  20  pack- 
ages, each  weighing  6.6QQ  lb.,  25  packages,  each  weighing  3.166 
lb.,  and  30  packages,  each  weighing  5.4  lb.  How  many  pounds 
and  ounces  do  the  three  shipments  together  weigh  ? 

41.  If  the  circumference  of  a  circle  is   10  yd.,  what  is  its 

diameter  in  feet  ?     (Circumference  3.1416  times  the  diameter.) 

42.  If  the  height  of  a  staircase  is  5  yd.,  and  that  of  each  step 
7Vs  inches,  how  many  steps  are  there  in  the  staircase  ? 

43.  An  astronomical  clock  lost  17.63  seconds  in  500  days. 
What  was  the  average  loss  per  day  ? 

44.  How  many  gallons  in  a  tank  11  ft.  deep,  7  ft.  long,  and 

6  ft.  wide  ? 

45.  What  will  1  foot  6  inches  cost,  if  5%  yards  cost  $2.10  ? 

46.  A  dairy-man  has  3  large  vessels  of  equal  capacity,  and  a 
small  one,  the  capacity  of  which  is  %  of  that  of  one  of  the  large 
ones.  The  3  large  vessels  together  hold  450  gal.  How  much 
will  he  pay  for  milk  to  fill  them  all  at  S1/^  a  qt.  ? 

47.  A  merchant  bought  4  cwt.  sugar  for  $38 ;  he  used  40  lb. 
himself,  and  sold  the  remainder  so  as  to  make  l1/^  profit  per  lb. 
on  the  whole  quantity.     How  much  per  lb.  did  he  sell  it  for  ? 

48.  Three  dozen  silver  spoons  weigh  1  lb.  2  oz. ;  how  much 
do  4  spoons  weigh  ?     (What  table  ?) 

49.  How  many  □  yds.  multiplied  by  17  will  equal  1530  □  ft.  ? 

50.  What  is  the  cost  of  2%  oz.,  if  %  lb.  cost  U<f  ? 


246  STANDARD  ARITHMETIC. 

51.  If  multiplying  a  number  of  feet  and  inches  by  3  and  by  4 
and  adding  the  products  give  76  ft.  5  in.,  what  is  the  number  ? 

52.  It  takes  4  men  75  days  of  8  h.  each  to  dig  over  a  certain 
piece  of  land  ;  how  many  hours  and  minutes  will  it  take  5  men  ? 

53.  A  stick  is  placed  perpendicularly,  so  that  1  yd.  27  in.  are 
above  ground.  It  throws  a  shadow  of  24/5  yards  in  length.  The 
shadow  of  a  church  steeple  near  by  is  at  the  same  time  of  the  day 
66  yd.  2  ft.  long.     How  tall  is  the  steeple  ? 

54.  A  man  earned  $1.25  every  week  day,  and  spent  $5.09 
every  week.     In  how  many  weeks  had  he  saved  $113.27  ? 

55.  Upon  the  circumference  of  a  wheel  are  48  teeth,  which 
are  exactly  1.294  inches  apart  (from  the  center  of  one  tooth  to 
the  center  of  another).     What  is  the  diameter  of  the  wheel  ? 

56.  If  1870  shingle  nails  weigh  8V8  lb.,  how  many  such  nails 
in  2  ounces  ? 

57.  A  locomotive  runs  90%  miles  in  2  h.  21  min. ;  at  the 
same  rate,  in  what  time  will  it  run  iy8  mile  ? 

58.  If  I  receive  $432  %  interest  in  .5  year,  how  much  do  I 
receive  in  4  mo.  ? 

59.  If  .3  yd.  are  equal  to  .9558  of  a  certain  measure,  how 
many  such  measures  are  equal  to  289  rd.  3  yd.  1  ft.  6  in.  ? 

60.  If  7/18  cu.  yd.  of  marble  costs  $18  %,  what  will  9  cu.  ft. 
cost? 

61.  To  travel  on  foot  from  A  to  B  in  6  days,  I  must  walk  2 
miles  an  hour  for  6  h.  every  day.  How  long  will  it  take  me  if  I 
make  3/4  of  a  mile  a  day  more  ? 

62.  In  3y3  years  the  interest  of  a  sum  of  money  is  $43  yB ; 
how  much  is  that  per  month  ? 

63.  Mr.  A  earns  $123  %  in  27  days  and  4  hours,  working  8 
hours  per  day.     What  is  that  per  day  ?    Per  hour  ? 

64.  A  type-setter  can  set  3/4  of  a  tabular  statement  in  53/4 
hours  ;  what  part  of  it  can  he  set  in  30  minutes  ? 


MEASURES.  247 

65.  The  light  of  the  sun  reaches  the  earth  in  8.22  min.,  thereby 
traveling  91242000  miles.  What  distance  does  it  travel  in  1 
second  ? 

66.  A  workman,  receiving  $%  per  hour,  was  paid  $13  %  for 
how  many  days  of  9  h.  each  ?    How  many  weeks  ? 

67.  A  locomotive  runs  270  miles  in  6.3  hours ;  how  many  miles 
is  that  per  hour  ?    Per  minute  ? 

68.  A  wheel  turns  35%  times  in  33  seconds;  how  often  in 
1  minute  ? 

69.  We  receive  the  oil  used  in  our  lamp  in  a  tin  can  8  in.  long 
and  wide  and  12  in.  deep.     How  much  oil  in  the  can  when  full  ? 

70.  A  railroad  track  rises  24%  ft.  in  a  distance  of  2086  ft. 
In  what  distance  does  the  elevation  amount  to  10  ft.  ? 

71.  The  wheel  of  a  wagon  in  turning  26  %  times  advances  319 
yd.  1  ft.  10  in.  What  is  the  circumference  of  the  wheel  ?  What 
is  its  diameter  ? 

72.  Mr.  A.  buys  3  casks  of  wine,  each  containing  52.04  gal., 
and  pays  $1.85  per  pt.     What  did  the  wine  cost  him  ? 

73.  How  many  T.,  cwt.,  and  lb.  in  a  block  of  marble  measur- 
ing 2  cu.  yd.,  if  %  cu.  yd.  weigh  2719.575  lb.? 

74.  A  grocer  received  a  quantity  of  coffee  in  3  bags,  the  first 
weighing  108  lb.  12  oz.,  the  second  96  lb,  8  oz.,  and  the  third 
120  lb.  2  oz.  The  cost  of  the  whole  was  $130.15.  What  was  the 
cost  per  cwt.  ? 

75.  The  fore-wheels  of  a  wagon  are  6  ft.  7.56  in.  in  circum- 
ference, the  hind- wheels  9  ft.  11.34  in.  How  often  do  the  latter 
turn  while  the  former  turn  108.6  times  ? 

76.  To  make  25  cwt.  of  bell-metal,  .8625  of  a  ton  of  copper 
and  .3875  of  a  ton  of  tin  are  required.  How  much  copper  and  tin 
are  required  for  a  bell  weighing  10  T.  ? 

77.  The  Thames  River  tunnel  measures  3960  yd.  in  length. 
How  many  meters  long  is  it,  35  yards  being  equal  to  32  meters  ? 


243  STANDARD  ARITHMETIC. 

78.  In  what  time  does  a  mason  build  63/4  cu.  yd.  of  masonry 
if  he  builds  at  the  rate  of  6.66%  cu.  yd.  in  5  days  ? 

79.  The  expenses  of  a  traveling  agent  amounted  to  $1012.50 
in  180  days  ;  how  much  were  they  per  day  ? 

80.  A  druggist  was  obliged  to  sell  40.5  lb.  of  drugs  for  the 
same  sum  for  which  he  had  bought  36.75  lb.  Receiving  only 
$9. 25  per  pound,  what  had  he  paid  per  oz.  ? 

81.  The  Cunard  steamer  Oregon  burns  on  an  average  337  tons 
of  coal  per  day,  the  ordinary  time  of  the  trip  being  6%  days. 
How  many  car-loads  of  coal  of  11  tons  each  are  required  for  one 
trip,  and  what  is  the  cost  of  the  coal  in  United  States  currency, 
at  11  shillings  per  ton  ? 

82.  How  many  chaldrons  of  charcoal  in  a  bin  15  ft.  long,  13 
ft.  6  in.  wide,  and  7  ft.  5  in.  deep  ? 

83.  How  many  gallons  in  a  cistern  8  ft.  in  diameter  and  7  ft. 
3  in.  deep,  if  the  capacity  for  each  10  inches  in  depth  is  313 
gallons  ? 

84.  What  is  the  difference  between  two  lots  of  land,  one  con- 
taining 23  square  rods,  the  other  being  23  rods  square  ?  (23  rods 
on  each  side.) 

85.  How  many  cubes  measuring  3  inches  each  way  can  be  cut 
from  a  cubic  yard  of  marble  ? 

86.  In  a  schoolroom  measuring  32  ft.  8  in.  long,  28  ft.  wide, 
and  14  ft.  6  in.  high,  how  many  cubic  feet  of  space  to  each  one 
of  56  pupils  ? 

87.  If  a  tank  is  5  ft.  6  in.  long,  and  3  ft.  wide,  how  much 
does  the  water  in  it  weigh,  when  the  water  is  2  ft.  9  in.  deep  ? 

88.  A  laborer  is  employed  to  saw  four  piles  of  cord-wood,  each 
12  ft.  long  and  5  ft.  8  in.  high,  at  75^  per  cord.  What  will  he 
receive  for  the  job  ? 

89.  A  bin  full  of  wheat  and  a  tank  full  of  water  each  measure 
5  ft.  8  in.  long,  4  ft.  wide,  and  3  ft.  9  in.  deep.  How  many 
quarts  does  one  contain  more  than  the  other  ? 


CHAPTER  XII. 

METRIC    SYSTEM    OF  WEIGHTS  AND    MEASURES. 

255.  The  metric  standard  for  the  measurement  of  distance  is 
the  Meter,  which  is  39.37  inches  long  (very  nearly  3  ft.  33/8  in.). 

256.  From  the  meter  all  other  measures  of  this  system  are 

derived,  hence  the  name  Metric  System. 

The  rule  represented  below  is  one  tenth  of  a  meter  in  length.     It  is  subdivided 
into  ten  equal  parts,  or  hundredths,  and  these  again  into  tlwusandtlis. 


One  decimeter  (tenth  of  a  meter)  subdivided  into  centimeters  (hundredths)  and 
millimeters  (thousandths). 

257.  The  capacity  of  a  box  one  tenth  of  a  meter  long,  wide, 
and  deep  (see  cut)  is  the  standard  unit  for  both  dry  and  liquid 
measures.  Such  a  measure  is  called  a  Liter  (pronounced  Lee'ter), 
and  is  equal  to  1.0567  liquid  quarts. 

258.  The  weight  of  so  much  pure  water  as  would  fill  a  meas- 
ure one  hundredth  of  a  meter  long,  wide,  and  deep  is  the  stand- 
ard unit  of  weight,  and  is  called  a  Gram.  A  gram  is  equal  to 
15.432  grains. 

Note. — To  familiarize  pupils  with  the  meter,  liter,  and  gram,  the  school  should 
be  supplied  with  a  meter-stick,  a  liter  measure,  and  a  gram  weight. 

In  absence  of  this  apparatus,  the  pupil  can  make  a  set  for  himself.  Let  him 
cut  a  stick  3  ft.  33/8  in.  long,  and  he  will  have  a  meter-stick.  Let  this  be  divided 
by  cross  lines  into  ten  equal  parts ;  these  will  be  decimeters  (tenths  of  a  meter — 
same  length  as  that  of  the  rule  represented  above).  Or,  he  can  make  a  pocket  meter 
out  of  a  piece  of  cord,  cut  long  enough  to  allow  for  tying  knots,  by  which  to  divide 
the  whole  into  10  equal  parts. 


250  STANDARD  ARITHMETIC. 

A  box  one  decimeter  long,  wide,  and  deep,  made  of  pasteboard  or  tin,  will  serve 
very  well  to  represent  the  liter,  which  is  the  unit  of  dry  and  liquid  measures.  The 
large  square  represented  on  page  256  is  of  the  same  size  that  the  bottom  and  sides 
of  the  box  should  be. 

If  the  box  were  made  of  tin,  and  filled  with  water  at  a  certain 
temperature  (39.2°  Fahrenheit),  the  water  would  weigh  a  thousand 
grams,  called  a  kilogram.  This  is  the  standard  for  weighing  gro- 
ceries, etc.  A  gram  would  be  the  weight  of  the  water  contained  in 
a  little  cup  that  might  be  molded  around  a  block,  the  size  of  which 
is  exactly  represented  by  the  cut  in  the  margin. 


EXERCISES. 

1.  With  a  meter-stick,  or  string  one  meter  in  length,  measure 
the  height  of  your  desk  ;  the  width  of  the  school-room  door ;  the 
length  and  width  of  the  room  ;  the  length  and  width  of  the 
school-house  ;  the  length  and  width  of  the  platform ;  of  a  win- 
dow ;  the  width  of  the  nearest  street  or  road. 

2.  Ascertain  how  many  steps  you  have  to  take  to  go  5  meters ; 
10  meters  ;  20  meters. 

3.  Ascertain  how  many  liters  there  are  in  a  peck  of  oats ;  in 
a  quart  of  beans  (dry  measure) ;  in  a  bushel  of  wheat.    (If  a  liter 

were  equal  to  a  quart  (dry  measure),  how  many  liters  should  a  bushel  hold?) 

4.  With  the  use  of  a  balance,  ascertain  the  common  weight  in 
grams  of  a  stick  of  candy  ;  of  a  slate  pencil ;  of  a  primer  or  a 
first  reader  ;  of  your  knife  ;  a  key  ;  a  rule,  etc. 

To  perform  most  calculations  in  metric  weights  and  measures  nothing  more  is 
needed  than  knowledge  of  decimals.  The  following  exercises  will  be  readily  per- 
formed without  further  explanation : 

5.  Add  35.6  m.,  456.35  m.,  93.12  m.,  6375.01  m.,  0.931  m. 

6.  Add  46.325  m.,  0.56  m.,  842.1  m.,  3.004  m.,  621.583  m. 

7.  From  563.83  m.  take  the  sum  of  98.375  m.  and  61.094  m. 

8.  From  832  liters  take  the  difference  between  156.22  1.  and 
2.345  1. 

9.  Find  the  cost  of  83.75  m.  of  cloth  at  $3.25  per  meter. 

10.  Find  the  cost  of  6.5  liters,  at  $1.85  per  liter. 


METRIC  SYSTEM  OF  WEIGHTS  AND  MEASURES.      251 

We  express  the  greater  distances  in  miles  and  rods,  and  smaller  ones  in  yards, 
feet,  and  inches ;  so  in  the  metric  system  the  higher  and  lower  denominations  are 
used  for  different  purposes. 

259.  In  the  following  table  the  names  of  the  orders  given 
above  the  line  of  dots  are  the  same  that  are  found  in  the  decimal 
numeration  table,  see  page  177.  The  names  below  the  dots  are 
the  corresponding  names  applied  to  the  metric  linear  measure  : 


Names  of  Orders 
used  in  Notation 
and  Numeration. 

Corresponding 

Names  applied  to 

Metric  Linear 

Measure. 


S 


fcq    f    ^    FS 

The  prime  unit  is  the  meter  =  39.37  inches.  Ten  units  of 
any  order  =  1  of  the  next  higher. 

260.  By  substituting  liter  for  meter  we  have  the  table  for 
measures  of  capacity,  and  by  substituting  gram  for  meter  we 
have  the  table  for  measures  of  weight. 

The  following  are  the  denominations  thus  formed  : 


Linear. 

Dry  and  Liquid. 

Weight 

Mil'li-me'ter 

(mm. ) 

Milli-li'ter  (ml.) 

Milli-gram 

(mg-) 

Cen'ti-meter 

(cm.) 

Centi-liter   (cl.) 

Centi-gram 

(eg.) 

Dec'H-meter 

(dm.) 

Deci-liter    (dl.) 

Deci-gram 

(dg.) 

Me'ter 

(m.) 

Liter         (1.) 

Gram 

(g.) 

Dek'a-meter 

(Dm.) 

Deka-liter  (Dl.) 

Deka-gram 

(Dg.) 

Hec'to-meter 

(Hm.) 

Hecto-liter  (HI.) 

Hecto-gram 

(Hg.) 

Kil'o-meter 

(Km.) 

Kilo-liter    (Kl.) 

Kilo-gram 

(Kg.) 

Myr'i-a-metei 

•(Mm.) 

Myria-liter  (Ml.) 

Myria-gram 

Quint'-al 

Ton-neau' 

(Mg.) 

(Q.) 

(T.) 

252 


STANDARD  ARITHMETIC. 


Notes. — 1.  Notice  that  the  abbreviations  for  measures  larger  than  the  unit 
begin  with  capital  letters,  the  abbreviations  for  measures  smaller  than  the  unit  be- 
gin with  small  letters.  Let  the  pupils  construct,  for  class  use,  oral  exercises  similar 
to  the  following,  on  all  the  tables,  using  the  appropriate  abbreviations. 

2.  The  names  of  the  denominations  may  be  readily  learned  by  repeating  only 
parts  of  the  names,  thus :  milli,  centi,  deci,  meter,  deka,  hekto,  kilo,  myriameter. 
As  these  are  repeated,  the  learner  should  think  of  the  meanings  of  the  prefixes, 
which  are  as  follows : 

The  Meanings  of  the  Prefixes. 


GREEK. 

SIGNJFICATION. 

LATIN. 

SIGNIFICATION. 

Deka- 

ten. 

Deci- 

tenth. 

Hekto- 

hundred. 

Centi- 

hundredth. 

Kilo- 

thousand. 

Milli- 

thousandth. 

Myria- 

ten  thousand. 

ORAL     EXERCISES. 

1.  How  many  meters  (m.)  in  a  myriameter  (Mm.)?  In  a 
kilometer  (Km.)  ?  In  a  hektometer  (Hm.)  ?  In  a  dekameter 
(Dm.)? 

(Myria  =  ten  thousand ;  kilo  =  a  thousand ;  hekto  =  a  hundred ;  delta  =  ten.) 

2.  How  many  dekameters  (Dm.)  in  a  hektometer  (Hm.)?  In 
a  kilometer  (Km.)  ?    In  a  myriameter  (Mm.)  ? 

3.  How  many  Aerometers  (Hm.)  in  a  myriameter  (Mm.)  ? 
In  a  kilometer  (Km.)  ? 

4.  How  many  Km.  in  a  Mm.  ? 

5.  What  part  of  a  m.  is  a  decimeter  (dm.)  ?  A  centimeter 
(cm.)  ?    A  millimeter  (mm.)  ? 

(Deci  =  one  tenth ;  centi  =  one  hundredth ;  milli  =  one  thousandth.  Compare 
decimal,  cent,  mill.) 

6.  How  many  mm.  make  one  cm.  ?     One  dm.  ?    One  m.  ? 

7.  How  many  cm.  make  one  dm.  ?     One  m.  ?     One  Dm.  ? 

8.  What  part  of  one  Km.  is  one  cm.  ?    Oue  Hm.  ?    One  mm.  ? 

9.  What  part  of  two  cm.  are  two  mm.  ?  Of  five  dm.  are  five 
cm.  ?    Of  seven  Hm.  are  seven  m.  ? 

10.  How  many  ml.  in  1.5  cl.  ?  .75  dl.  ? 

11.  How  many  grams  in  .25  Mg.  ?    %  Dg.  ? 


METRIC  SYSTEM  OF  WEIGHTS  AND  MEASURES.     253 


SLAT  E     EXERCISES. 

l.  Write  the  Linear  Measure  table,  thus  : 


Table. 


10  millimeters 
10  centimeters 


1  centimeter. 
1  decimeter,  etc. 


2.  Write  the  Dry  and  Liquid  Measure  table  in  the  same  way. 

(The  table  of  Liters.) 

3.  Also  the  table  of  Weights.     (The  table  of  Grams.) 

4.  Write  in  full  the  denominations  indicated  by  m.,  cl.,  Dg., 
dm.,  Kl.,  cm.,  Mm.,  HI.,  mm.,  Mg.,  Dm.,  eg.,  ml.,  Kg.,  Hm., 
Dl.,  mg.,  Km.,  Hg.,  dl.,  g.,  dg.,  1. 

5.  Read  858.65  m.,  giving  separately  the  denomination  of  each 
figure.  Ans.,  8  hektometers,  5  dekameters,  8  meters,  6  deci- 
meters, 5  centimeters. 

6.  In  the  same  way  read  : 

89367.351  m.  62354.319  g. 

5432.019  g.  3124.009  m. 

10000.01    1.  85492.88    g. 

654.321  m.  987.002  1. 

30.50    g.  124.03    m. 

123.456  g.  98765.432  1. 

7.  Write  3  Kg.  5  g.  3  eg.  4  mg.  in  the  denomination  of  the 
prime  unit  (grams).     Ans.,  3005.034  g. 

Note. — Be  careful  to  fill  all  intervening  vacant  orders  with  ciphers,  so  that  each 
digit  shall  by  its  position  indicate  its  denomination.  In  the  number  given  above, 
there  are  no  hektograms,  no  dekagrams,  no  decigrams,  hence  the  ciphers. 


295.31    in. 

65.12    1. 

30.02    1. 

.203  1. 

3.892  g. 

18.303  ra. 

1.993  ra. 

33.5      g. 

384.002  g. 

8.654  1. 

50.023  1. 

.009  m. 

8.  In  the  same  way  write  : 

5  HI.  7  Dl.  8  dl.  2  cl.  5  ml. 

1  Mm.  6  Hm.  5  m.  3  mm. 

2  Dg.  5  g.  3  eg.  8  mg. 

6  Kl.  5  dl.  4  cl.  3  ml. 

4  Hm.  5  Dm.  3  dm.  8  mm. 

3  Kg.  8  dg.  5  eg.  7  mg. 


7  Dg.  3  g.  9  eg.  1  mg. 

9  Kl.  8  Dl.  2  1.  3  dl. 

5  Km.  6  Hm.  1  m.  3  dm. 

7  Mg.  2  Dg.  3  eg.  9  mg. 

7  Ml.  8  HI.  9  1.  1  cl. 

5  Mm.  9  Km.  5  Hm.  8  Dm. 


254  STANDARD  ARITHMETIC. 

Reductions. 

1.  Seduce  25.325  kilometers  to  decimeters. 

1  Km.  =  10  Hm.,  hence  25.325  Km.  =  253.25  Hm. 
1  Hm.  =  10  Dm.,  "  253.25  Hm.  =  2532.5  Dm. 
1  Dm.  =  10  m.,  "     2532.5  Dm.  =  25325.  m. 

1  m.     =10  dm.,        "      25325.  m.      =  253250.  dm. 
Thus,  in  Reduction  Descending,  each  step  removes  the  deci- 
mal point  one  place  to  the  right. 

261.  Mule, — To  reduce  a  higher  metric  denomination  to  a 
lower :  Remove  the  decimal  point  one  place  to  the  right  for  each 
step  of  the  reduction.    Annex  ciphers,  if  necessary. 

2.  Reduce  435.32  dm.  to  Hm. 

10  dm.  =  1  m.,  hence  435.32  dm.  =  43.532  m. 
10  m.    =lDm.,    <•'    43.532  m.     =  4.3532  Dm. 
10Dm.=  lHm.,    "    4.3532  Dm.  =  .43532  Hm. 
Thus,  in  Reduction  Ascending,  each  step  removes  the  decimal 
point  one  place  to  the  left. 

262.  Rule.  —  To  reduce  a  lower  metric  denomination  to  a 
higher:  Remove  the  decimal  point  one  place  to  the  left  for  each 
step  of  the  reduction.    Prefix  ciphers  if  necessary. 

The  two  foregoing  rules  may  be  given  in  one,  thus  : 

263.  Rule. — To  reduce  a  number  from  one  metric  denomina- 
tion to  another :  Remove  the  separatrix  from  the  right  of  the 
given  denomination  to  the  right  of  the  denomination  required, 
and  change  the  abbreviation  accordingly. 


EXERCISES. 

1.  How  many  cm.  in  7  m.  ?    How  many  dg.  in  3  Dg.  ? 

2.  How  many  meters  in  3.15  Km.?    How  many  liters  in  6.17 
HI.  ?    How  many  grams  in  18.416  mg.  ? 

3.  Express  the  sum  of  231  cm.,  2859  dm.,  354  mm.  in  meters. 

4.  Add  28.35  m.,  200.03  m.,  123.9  m.,  456.7  m.,  and  express 
the  answer  in  hectometers. 


METRIC  SYSTEM  OF  WEIGHTS  AND  MEASURES.     255 

5.  Express  in  grams  and  add  127  dg.,  7200  Hg.,  8. 83- Kg. 

6.  Find  how  many  grams  remain  if  you  take  8  Kg.  from 
58  Kg. 

7.  Carl  is  told  to  measure  the  water  in  a  vessel  containing 
14.31  1.,  with  a  cup  holding  3  cl.  How  often  will  he  have  filled 
the  cup,  if  his  measurement  is  correct  ? 

8.  The  circumference  of  May's  hoop  measures  3.8  m.;  how 
many  times  will  it  turn  in  rolling  a  distance  of  53.2  m.  ? 

9.  A  nickel  5^  piece  weighs  5  g.  How  many  such  pieces  can 
be  made  of  a  bar  of  coin  metal  weighing  5  kilograms. 

10.  Add  4.97  m.,  21  cm.,  6.03  m.,  9.137  m.,  38  dm. 

11.  Subtract  9  Km.  6  Dm.  7  m.  3  dm.  from  1  Mm. 

12.  Multiply  18.28  Dg.  by  29.     Express  the  product  in  eg. 

13.  Divide  5238.45  1.  by  8.     Express  the  quotient  in  Dl. 

14.  Sold  14.23  1.  at  $.50  a  dl.     How  much  did  I  receive  ? 

15.  A  train  runs  54.5  Km.  an  hour.     How  far  in  4.5  hours  ? 

16.  How  many  kilometers  of  telegraph  wire  are  needed  to  con- 
nect two  stations,  if  the  distance  between  two  poles  is  43  m., 
and  there  are  516  poles  between  the  stations  ?    (517  x  43.) 

17.  How  many  m.  of  fence  are  needed  to  close  in  a  field 
535.5  m.  long  and  285.5  m.  wide  ? 

18.  What  will  be  the  profit  on  1  %  HI.  of  vinegar,  bought  at 
U  a  HI.  and  sold  at  8^  per  liter  ? 

19.  What  will  be  the  profit  on  10  g.  of  calomel  bought  for  50^, 
if  sold  in  powders  of  5  dg.  at  5<f>  each  ? 

20.  A  merchant  bought  cloth  at  $1.14  a  m.;  for  how  much 
per  m.  must  he  sell  it  to  gain  1/3  of  the  cost  ? 

21.  If  I  buy  6.328  Ml.  at  5^  a  1.,  and  sell  it  at  2<f  a  dl.,  do  I 
lose  or  gain  ?    How  much  ? 

22.  How  many  1.  of  grain  will  fill  a  box  7  m.  long,  wide,  and 
deep  ? 


256 


STANDARD  ARITHMETIC. 


Square  Measure. 

Table. 
100  Sq.  mm.  =  1  Sq.  cm. 
100  Sq.  cm.   =  1  Sq.  dm. 
100  Sq.  dm.  =  1  Sq.  m. 
100  Sq.  m.     =  1  Sq.  Dm.  =  1  Are. 
100  Sq.  Dm.  =  1  Sq.  Hm. 
100  Sq.  Hm.  =  1  Sq.  Km. 

Notes. — 1.  For  land  measure,  the  square  Dm.  is  called  an  Are  (pronounced  like 
the  verb  are),  from  the  Latin  area,  which  means  a  level  piece  of  ground. 


This 

lame 

saua 

re    is 

0 

s< 

tl 

:mare     decimeter.      It    co 

iins     100     square     centim 

irs   or   10,000   square  mill 

leters. 

When    the   liter  measu 

made  in  cubic  form,  eac 

de   is   equal   to   this   entr 

^uare. 

One  hundred  squares  lil 

e- 

it 

n 

i- 

re 

is 

si 

h 

re 

S( 

4-1 

:e 

lis  equ 

a j.  uiie 

ueu  ta.it 

'■"■"■"■"l 

METRIC  SYSTEM  OF  WEIGHTS  AND  MEASURES.     257 

2.  The  standard  unit  of  land  measure  is  derived  directly  from  the  dekameter, 
instead  of  from  the  meter,  because  a  large  unit  is  more  convenient  for  measuring 
large  surfaces. 

The  pupil  may  provide  himself  with  a  string,  10  meters  long,  and  with  this 
measure  off  a  square  Dm.  on  the  schoolroom  floor,  or  on  the  play-ground.  Thus 
he  will  form  an  idea  of  the  French  measure  of  land,  and  measuring  off  1  □  m.  in  a 
corner  of  an  are,  he  will  have  a  ccntare.     The  table  for  land  measure  is 

100  Centares  (ca.)  =  1  Are  (a.). 

100  Ares  (a.)  =  1  Hectare  (Ha.). 

Inasmuch  as  in  square  measure  100  units  of  each  order  make  one  unit  of  the 
next  higher,  each  denomination  must  have  two  places.  For  the  same  reason,  cents 
have  two  places  in  writing  dollars  and  cents. 

Compare  5  dollars  7  cents,  which  is  written  $5.07,  with  5  a  Hm.  17  D  Dm.  6 
□  m.  5  a  dm.,  written  thus:  51706.05  a  m. 


EXERCISES. 

1.  Write  the  following  as  square  meters  : 

7  □  Km.  19  d  Hm.  30  □  Dm.  5  □  m.     Ans.,  7193005  □  m. 
5    o    Hm.    5    □   Dm.    5    □    m.   5    □    dm.   5    □   cm.     Ans., 
50505.0505  □  m. 

2.  Write  the  following  as  square  dm. :  6   □  Dm.  3   □   m.  81 
□  dm.  53  □  cm. 

3.  How  many  centares  are  19  hectares  59  ares  and  48  centares  ? 

4.  Express  in  ares  58  a  Km.  93  a  Hm.  73  a  Dm.  42  □  dm. 

5.  How  many  □  m.  in  2509703  □  mm.  ?    In  35020509  □  mm.  ? 

6.  Reduce  17.519  □  m.  to  □  cm.;  also  to  □  mm. 

7.  If  V«  of  a  □  m.  costs  $14.90,  what  will  290  □  cm.  cost  ? 

8.  Find  the  area  of  a  floor  5.2  m.  long,  and  3.6  m.  wide. 

9.  How  many  bricks,  each  20  cm.  long  and  10  cm.  wide,  will 
it  take  to  pave  a  cellar  10  m.  long  and  8.5  m.  wide  ? 

10.  Reduce  2.1736  Ha.  to  □  m.;  517.3  centares  to  □  m. 

11.  Reduce  3872847  □  m.  to  a.;  also  to  Ha. 

12.  How  many  hectares  in  a  lot  59  m.  long  and  21  m.  wide  ? 

13.  Reduce  12856  ares  to  hectares  ? 


258 


STANDARD  ARITHMETIC. 


14.  Find  tne  area  of  your  schoolroom  in  metric  measurement. 

15.  Supposing  your  school-lot  to  be  a  rectangle  140.5  m.  long 
and  70.5  m.  wide,  and  the  buildings  to  occupy  just  400  □  m., 
what  space  is  left  for  play-ground  ? 

16.  Express  the  result  of  Ex.  15  in  d  m. ;  in  centares. 

17.  Find  how  many  rolls  of  wall-paper,  10  m.  long,  1/2  a  meter 
wide  are  needed  to  paper  a  room,  the  size  of  which  is  6.5  m.  X  4.2 
m.  X  3.2  m.     Deduct  2.47  □  m.  for  window  and  door. 

18.  Mr.  Quinn  had  5  hectares,  5  ares,  9  centares  of  land,  and 
sold  first  0.5,  then  0.3  of  it  for  $384  an  are.  What  did  he  get  for 
what  he  sold  ?     How  much  was  left  ? 


Cubic  Measure. 

Table. 
1000  Cu.  mm.  =  1  Cu.  cm. 
1000  Cu.  cm.   =  1  Cu.  dm.  ==  1  Liter  (capacity). 
1000  Cu.  dm.  =  1  Cu.  m.    =  1  Stere  (pronounced  stair). 


METRIC  SYSTEM  OF  WEIGHTS  AND  MEASURES.     259 

Wood   Measure. 
10  Decisteres  (ds.)  =  1  Stere  (s.). 
10  Steres  (s.)  =  1  Dekastere  (Ds.). 

Notes. — 1.  From  the  cut  on  page  258  the  pupil  may  see  that  a  cu.  m.  (=  1 
stere)  is  equal  to  1000  cu.  dm.,  and  that  1  cu.  dm.  (=1  liter)  is  equal  to  1000  cu. 
cm.  (milliliters).  If  the  cut  were  twice  as  long,  wide,  and  high,  it  would  represent 
a  liter  of  actual  size. 

2.  From  the  same  cut  (which  is  only  1/20  of  a  meter  each  way)  the  pupil  can 
see  that  a  cu.  m.  is  100  times  100  times  100  cu.  cm ,  or,  in  other  words,  that  a 
cubic  meter  contains  1  million  cubic  centimeters. 

264.  Since,  in  cubic  measure,  1000  units  of  each  denomina- 
tion make  one  unit  of  the  next  higher  order,  each  denomination 
must  have  three  places.  For  instance  :  3  cu.  dm.  would  be 
written  0.003  ;  3  cu.  cm.  would  be  written  0.000003  ;  3  cu.  mil- 
limeters would  be  written  0.000,000,003. 


EXERCISES. 

1.  Write  319  cu.  m.  99  cu.  dm.  285  cu.  cm.  and  4  cu.  mm. 
Arts.,  319.099285004  cu.  m. 

2.  Express  in  steres,  19  dekasteres  6  steres  7  decisteres. 

Note. — The  units  of  wood  measure  form  a  scale  of  tens ;  each  denomination, 
therefore,  needs  but  one  place. 

3.  Reduce  7  Ds.  5  s.  and  6  ds.  to  ds. 

4.  Reduce  29  cu.  m.  312  cu.  dm.  703  cu.  cm.  to  cu.  dm. 

5.  Add  3  cu.  m.  18  cu.  dm.  207  cu.  cm.;  385  cu.  m.  230  cu. 
dm.  895  cu.  cm.  10  cu.  mm.;  831  cu.  m.  300  cu.  cm.  Express 
the  sum  in  cu.  meters,  then  in  cu.  dm. 

6.  How  many  cu.  m.  of  earth  must  be  removed  to  build  a  cis- 
tern 3.5  m.  deep,  and  1.8  m.  wide  both  ways  ? 

7.  I  had  to  pay  $1.50  per  cu.  m.  for  excavating  and  removing 
earth.  The  hole  made  was  6.5  m.  long,  5.2  m.  wide,  3.3  m.  deep. 
How  much  did  I  have  to  pay  ? 

8.  How  long  must  a  pile  of  wood  be  so  that  it  may  contain  13 
steres,  if  it  is  4.5  m.  high  and  2.3  m.  wide  ? 


260 


STANDARD  ARITHMETIC. 


9.  How  many  loads  of  earth,  each  filling  3.25  cu.  m.,  will  fill 
a  hole  12.3  m.  long,  C.5  m.  wide,  and  5.1  m.  deep  ? 

10.  What  is  the  cost  of  building  a  wall  of  masonry  2.3  m. 
high,  17.65  m.  long,  and  .35  m.  thick,  at  $7.45  a  cu.  m.? 


Tables  of  Equivalents. 


1  mm.  = 
1  cm.  = 
ldm.  = 
1  m.  = 
lDm.= 
lHm.= 
lKm.= 


Measures  of  Lenglh. 

0.03937  of  an  Inch. 
0.3937  of  an  inch. 


3.937 


393.7 
3937. 
39370. 


1  Mm.  =  393  700. 


inches, 
inches, 
inches, 
inches, 
inches, 
inches. 


Measures  of  Capacity. 
1  ml.  =     0.0010567  of  a  qt.  (Liquid) 
lcl.  =     0.010567  ofaqt. 
ldl.  =     0.10567     ofaqt. 
11.     =     1.0567       quarts 
1D1.  =       .28375     ofabu.  (Dry), 
1H1.=     2.8375      bushels     " 
1K1.=  28.375         bushels    " 
1  Ml.  =283. 75  bushels    " 


Measures  of  Weight. 

1  mg.  =  0.015432  of  a  grain. 

leg.  =  0.15432    of  a  grain, 

ldg.  =  1.5432      grains. 

1  g.  =  15.432        grains. 

1  Dg.  =  0.022046  of  a  lb.  (Avoir.). 

lHg.  =  0.22046  of  a  lb. 

lKg.  =  2.2046     lb. 

lMg.  =  22.046       lb. 

1  Quintal  =  220.46         lb. 


1  Tonneau  =  2204.6 


lb. 


Square  Measure. 
1  d  cm.      =      0.1550  of  a  n  inch. 
1  d  dm.     =    15.50       □  inches, 
lam.        =      1.196     □  yd. 
1  are  =  119.6        □  yd. 

1  hectare  =      2.471  acres. 

Cubic  Measure 
1  cu.  cm.  =    0.061  cu.  in. 
1  cu.  dm.  =  61.027  cu.  in. 
1  cu.  m.,  or  1  stere  =    1.3079  cu.  yd.,  or  0.2759  of  a  cord. 


METRIC  SYSTEM  OF  WEIGHTS  AND  MEASURES.    261 

SLAT  E    EXERCISES. 

1.  How  many  feet  in  9  dm.?    In  682  mm.? 

2.  How  many  pounds  av.  in  2000  kilos  ?    (Kilo  is  the  commercial 
name  for  kilogram.) 

3.  One  lb.  av.  is  what  decimal  fraction  of  one  kilo  ? 

4.  10  cords  equal  how  many  steres  ? 

5.  How  many  gallons  in  20  liters  ? 

6.  How  many  hectares  in  2471  acres  ? 

7.  5678  bushels  equal  how  many  hektoliters  ? 

8.  One  common  ton  is  what  part  of  a  metric  ton  ? 

9.  I  imported  1000  m.  silk  at  a  cost  of  12  francs  per  m\3ter, 
and  sold  it  at  $2.50  per  yard.     How  much  did  I  gain  ? 

10.  A  grocer  bought  10  HI.  of  potatoes  at  $1.00  per  hektoliter, 
and  sold  them  at  $.50  per  bu.    Did  he  gain  or  lose,  and.  how  much  ? 


Original   Problems. 

1.  Ask  certain  members  of  the  class  to  bring  to  school  a  meter 
stick,  or  a  cord  or  tape-line  a  meter  long,  marked  off  into  deci- 
meters, and,  if  possible,  into  centimeters;  ask  others  to  bring  a 
liter  measure,  others  a  kilogram  of  lead  or  nails,  others  a  sheet 
of  paper  measuring  a  centare.  Require  that  the  measures  shall 
be  made  by  the  individuals  who  bring  them. 

2.  Ask  for  a  description  of  the  are  and  of  the  stere. 

3.  Give  the  measurements  made  in  feet  and  inches  by  yourself 
of  some  known  object  of  moderate  length,  say  of  the  fence  on  one 
side  of  the  school-yard,  and  ask  the  members  of  the  class  to  com- 
pute the  measure  in  meters.  Then  test  by  the  use  of  the  most 
accurate  metric  measure  you  can  get.  Apply  this  method  to 
measures  of  capacity,  of  surface,  of  solids,  and  of  weight. 

4.  Ask  each  member  of  the  class  to  report  his  weight  in  kilo- 
grams, having  first  found  it  in  pounds,  and  require  the  others  to 

reduce  the  kilograms  to  avoirdupois  weight, 
i  it 


CHAPTER    XIII. 

PRACTICAL    MEASUREMENTS. 

Lumber. 

265.  Boards  1  inch  or  less  in  thickness  are  estimated  by  the 
square  foot. 

Thus,  a  board  16  feet  long,  12  inches  wide,  and  1  inch  or  less  in  thickness  would 
contain  16  n  feet.  A  board  16  feet  long,  11  inches  wide,  1  inch  thick  or  less,  would 
contain  lx/i2  of  16  □  feet  =  142/3  □  feet,  etc. 

266.  When  lumber  is  more  than  1  inch  thick  the  thickness 
is  taken  into  account,  and  the  board  foot,  1  foot  square  and  1 
inch  thick,  becomes  the  standard  by  which  it  is  estimated. 

Thus,  a  piece  of  lumber  16  ft.  long  12  in.  wide  and  1  1/4  in.  thick  would  con- 
tain 16  board  feet  +  J/4  of  16  =  20  board  feet.  If  1  */,  in-  thick  it  would  con- 
tain 24  board  ft.,  if  2  in.  thick  it  would  contain  32  board  feet,  etc.  A  plank  2 
inches  thick  is  thus  reckoned  as  two  boards,  each  an  inch  thick. 

12  board  feet  =■  1  cubic  foot. 

In  the  measurement  of  the  width  of  a  board  a  fraction  greater  than  a  half  inch 
is  called  a  half,  and  if  less  than  a  half  it  is  rejected. 

The  width  of  a  tapering  board  is  measured  at  the  middle,  or  half  the  sum  of 
the  end  measurements  is  taken  as  the  mean  or  average  width. 

267.  Hewn  timber  is  sold  either  by  board  or  cubic  measure. 

l.  Find  the  cost  of  boards  16  ft.  long,  1  in.  thick,  of  dif- 
ferent widths,  as  below,  @  $31  per  M ;  also  their  value  if  18  ft. 
long: 

9*/f  in. 
10Vam. 
llVtin. 

iM/i'to. 

Suggestion.— If  the  boards  were  laid  side  by  side,  how  many  square  feet  would 
they  cover  ?     Only  one  multiplication  is  needed. 


12  in. 

16  in. 

15  in. 

19*/fra. 

19  in. 

13  in. 

14Vain. 

18  in. 

17V2in. 

18Vt  in. 

11  in. 

12V2in. 

14  in. 

8  in. 

17  in. 

9  in. 

2Q  in. 

10  in. 

7  in. 

16  in. 

PRACTICAL  MEASUREMENTS.  263 

2.  How  many  board  feet  in  a  stick  of  timber  15  in.  wide,  14 
in.  thick  and  20  ft.  long  ?    How  many  cubic  feet  ? 

Analysis. — 1.  A  board  1  in.  thick, 

Solution.  15  in.  wide,  and  1  ft.  long,  would  contain 

15  /         300  /       _  ok  k  *f  x  5/i  2  of  a  board  foot,  and  if  20  ft.  long 

20  X       1 12—       i  ]2  —  25  board  It.         ft  wouM  contain  20  x  15/12  =  25  board 

14  x  25  =  350  board  ft.  feet    But  a  Piece  of  tfanbf  "  *- thick 

contains  14  times  as  much  lumber  as  a 

board  of  the  same  length  and  width 

350  -J-  12  =  29  Ve  cubic  ft.  an(i  on]y  i  m.  thick.     14  x  25  =  350 

board  feet. 

2.  Twelve  board  feet  being  equal  to  a  cubic  foot,  350  board  feet  contain  as 
many  cubic  feet  as  there  are  times  12  board  feet  in  350,  which  is  29  1/6.  Hence  in 
350  board  feet  there  are  29  1/e   cubic  feet.     Am. 

3.  How  many  board  feet  in  29  joists,  each  28  ft.  long,  16  in. 
wide,  and  3  in.  thick  ? 

4.  At  $25  per  M,  what  will  be  the  cost  of  8  inch  square  tim- 
bers, measuring  respectively  18,  24,  22,  16,  32,  and  28  feet  long  ? 


Masonry  and  Brick  Work. 

268.  Masonry  is  commonly  estimated  by  the  perch. 

A  perch  of  masonry  actually  contains  24  3/4  cubic  feet,  its  dimensions  being 
16  1/2  x  1 1/2  x  1  ft.,  but  it  is  variously  estimated  in  different  localities,  sometimes 
at  only  16  1/2  cubic  feet.  It  is  gradually  falling  into  disuse,  and  the  cubic  foot  and 
yard  taking  its  place. 

1.  Find  how  many  perches  of  masonry  in  the  walls  of  a  cellar, 
that  is  50  ft.  long  and  43  ft.  wide,  the  walls  being  8  ft.  high 
and  24  in.  thick,  112  cubic  feet  being  allowed  for  openings. 

In  estimating  material,  corners  are  measured  once,  and  allowance  is  made 
for  doors  and  windows.  In  estimating  labor,  the  corners  are  measured  twice, 
and  usually  only  J/2  *s  deducted  for  openings. 

2.  How  much  will  the  bricks  for  a  wall  45  ft.  long,  7  ft.  high, 
and  16  in.  thick  cost  at  $8.75  per  M,  one  gate-way  4  ft.  wide 
being  deducted. 

Fourteen  common  bricks  are  usually  allowed  to  a  d  foot  in  one  face  or  side  of 
an  8  in.  wall,  and  7  additional  bricks  for  every  4  in.  increase  in  width.  Bricks 
nominally  of  the  same  style  and  from  the  same  manufacturer  vary  in  size,  so  that  a 
table  of  exact  dimensions  is  impracticable. 


264 


STANDARD  ARITHMETIC. 


Flooring. 

1.  Find  the  cost  of  flooring  the  parlor,  library,  sitting  and 
reception  rooms  with  lumber  @  $35  per  M,  making  no  allowance 
for  waste — the  cost  of  laying  the  floor  being  $1.50  per  square. 

Note. — 100  square  feet  of  surface  is  called  a  square. 

Suggestion. — Find  first  how  many  feet  of  lumber  will  be  needed  and  what  it 
will  cost,  and  then  the  cost  of  laying  the  floor  at  the  given  price  per  square. 

2.  Find  the  cost  of  flooring  the  dining-room  with  3  in.  ash 
@  $45  per  M,  no  allowance  for  waste,  and  the  cost  of  laying 
being  I2.121/,  per  square. 

3.  Find  the  cost,  @  2<f  per  board  foot,  of  flooring  joists,  8  in. 
wide,  21/2  in.  thick,  and  of  various  lengths,  as  follows  : 

For  hallway,  34  joists,    8  ft.  8  in.  For  reception-room,  15  joists,  17  It. 

"    parlor,     22      u       14  u  8  u  "    sitting-room,       22      "         " 

"    library,    15      "       12  "  4  "  "    dining-room,       21      "         " 


PRACTICAL  MEASUREMENTS.  265 

Plastering. 

269.  The  processes  of  calculating  the  cost  of  plastering  and  painting  are  quite 
simple,  but  the  rules  for  the  measurement  of  the  work  vary  in  different  localities, 
and  require  experience  in  their  application.  It  is  held  by  some  authorities  to  be  an 
equitable  rule  for  plain  work  to  measure  all  the  walls  and  ceilings  without  deduct- 
ing anything  for  an  opening  of  less  extent  than  1  superficial  yards  (63  a  feet). 

Find  the  cost  of  plastering 

1.  The  rooms,  they  being  of  the  uniform  height  of  11  ft.,  @ 

400  a  □  yd.      No  allowance  for  openings. 

2.  What  would  be  the  cost  of  the  same  work  at  the  same  price 
per  □  yd.  if  allowance  be  made  for  doorways  and  windows,  their 
dimensions  being  as  follows  : 

Doorways — Front,  5  ft.  by  9  ft. ;  the  doors  between  parlor  and  library,  and 
between  reception-room  and  sitting-room,  each  6  ft.  by  9  ft. ;  all  others,  3  ft.  by 
8  ft. 

Windoics— Front  windows,  3  ft.  4  in.  by  9  ft. ;  all  others,  3  ft.  4  in.  by  1  ft.  8  in. 


Painting  and  Kalsomining. 

Find  the  total  cost  of  painting 

1.  The  base-boards  of  the  rooms.  They  are  9  in.  wide,  and 
require  3  coats  of  paint  @  100  a  □  yd.  per  coat.     (Deduct  width  of 

doorways.) 

In  practice,  surfaces  less  than  6  inches  wide  are  measured  as  6,  and  if  more 
than  6  and  less  than  12  inches  wide,  are  measured  as  12.  The  cost  of  painting 
is  here  computed  accordingly. 

2.  The  doors  and  windows,  both  sides,  at  550  a  □  yard,  the 
number  and  dimensions  of  which  are  given  above. 

In  finding  the  cost  of  painting  doors  it  is  customary  to  add  one  edge  to  each 
side,  but  here  the  dimensions  may  be  used  as  given.  The  cost  of  painting  the  win- 
dows may  be  found  as  if  they  were  plain  surfaces  of  same  dimensions.  This  is  a 
very  common  rule  when  there  are  more  than  two  lights  in  the  window. 

3.  The  library  and  dining-room  floors  @  280  a  o  yard. 

4.  The  outside,  measuring  18  ft.  by  196  ft.,  2  coats,  each 
9^  a  □  yd.,  deducting  doors  and  windows. 

5.  Find  the  cost  of  kalsomining  the  ceilings  (including  hall) 
@  50  a  □  yd. 


266  STANDARD  ARITHMETIC. 

Paper  Hanging. 

270.  Wall  paper  is  sold  only  by  the  roll,  any  part  of  a  roll  being  counted  as 
a  whole  one. 

American  paper  has  8  yd.  in  a  roll,  and  is  commonly  18  in.  wide.  (Foreign 
papers  differ  as  to  width  and  length  of  roll.)  Borders  are  sold  by  the  yard,  and 
vary  in  width  from  3  in.  to  18  in.     A  very  wide  border  is  called  a  friese. 

The  exact  cost  of  papering  a  room  can  be  ascertained  only  by  taking  account 
of  the  number  of  rolls  actually  used  in  doing  the  work,  but  it.  is  useful  to  be  able 
to  make  an  approximate  estimate,  which  may  be  done  as  follows :  Find  the  distance 
around  the  room,  omitting  all  openings.  Divide  the  number  of  half  yards  thus 
found  by  the  number  of  entire  strips  that  can  be  cut  from  a  roll,  that  is,  by  2,  if 
the  height  from  baseboard  to  ceiling  or  cornice  is  more  than  8  and  less  than  12  ft., 
or  by  3,  if  the  height  is  not  more  than  8  ft. 

When  the  length  of  the  strips  is  such  as  to  leave  much  waste  in  cutting  a  roll, 
a  double  roll  (16  yd.)  can  often  be  used  to  better  advantage,  thus :  If  the  length  of 
a  strip  be  9  ft.  6  in.,  a  single  roll  will  make  two  strips  with  5  ft.  waste,  while  a 
double  roll  will  make  5  strips  with  only  6  in.  waste. 

When  the  paper  cuts  with  little  or  no  waste,  an  additional  roll  or  two  will  be 
required  for  the  spaces  under  and  over  the  openings. 

l.  Find  the  number  of  rolls  of  paper,  8  yards  long  and  18  in. 
wide,  required  for  each  room,  all  being  of  the  uniform  height  of 
11  ft. ,  allowing  for  doors  and  windows  according  to  the  widths 
given  on  page  265.  

Carpeting. 

271.  The  number  of  yards  of  carpeting  needed  for  any  given  room  can  not 
always  be  ascertained  by  calculating  the  number  of  square  feet  or  yards  in  the  floor, 
for  unless  either  the  length  or  width  of  the  room  is  a  multiple  of  the  width  of  the 
carpeting,  and,  furthermore,  unless  the  carpet  will  match  in  the  length  required, 
more  or  less  will  have  to  be  turned  under  or  cut  off  at  the  end  or  side,  and  some- 
times both. 

A  carpet  with  small  figures  will  generally  lose  less  in  matching  than  one  with 
large  ones.  Care  should  be  taken  to  lay  the  carpet  so  as  to  lose  as  little  as  possible 
either  in  matching  or  in  width. 

Find  the  cost  of  carpeting 

l.  The  parlor  with  Brussels  carpet,  27  in.  wide,  @  $1.50,  to 
be  laid  lengthwise,  3  in.  being  lost  on  each  strip  to  match.  If 
the  floor  is  first  covered  with  paper-lining,  @  9$  per  □  yard, 
what  will  be  the  additional  cost  ? 


PRACTICAL  MEASUREMENTS.  267 

2.  The  sitting-room  with  yard-wide  three  ply  carpet,  laid 
lengthwise,  @  95^,  3  inches  on  each  strip  being  lost  to  match. 

3.  The  library  with  China  matting,  36  in.  wide,  @  60^,  there 

being  no  loss  to  match.      (Allow  1  y2  in.  at  each  end  of  each  strip  for  turn- 
ing in.) 

4.  The  dining-room  with  China  matting,  36  in.  wide,  @  50^, 

to  be  laid  with  least  loss,  no  loss  to  match.      (Three  in.  on  each  strip 
being  allowed  for  turning  in,  as  above.) 

5.  The  reception-room  with  Brussels  carpet,  27  in.  wide,  % 
$1.60,  to  be  laid  lengthwise,  6  in.  being  lost  on  each  breadth  to 
match. 

6.  The  hall  with  Moquette  carpet,  27  in.  wide,  @  $1.75. 

7.  What  will  be  the  cost  of  a  rug  for  the  dining-room,  that  is 
3  yd.  1  ft.  6  in.  by  4  yd.,  @  85^  per  □  yard,  with  a  border  in 

addition,  @  65^  per  lineal  yard  ?     (Allow  1  yard  of  border  for  turning 
corners.) 

8.  Find  the  cost  of  carpeting  a  stairway  having  20  steps,  and 
each  one  having  11%  in.  tread  and  6%  in.  rise,  %  yard  being 
allowed  for  the  landing,  1%  yard  for  the  turning,  and  %  yard 
for  moving  up  when  the  edges  are  worn.     Carpet,  $1.50  per  yard. 


Paving. 

Find  the  cost  of  paving 

1.  A  sidewalk  4  ft.  wide  and  63  ft.  long,  @  21  ^  per  □  foot. 

2.  A  courtyard  19  ft.  X  19  ft.  6  in.  with  brick  laid  flat  in 
sand,  @  75^  a  □  yard. 

3.  A  sidewalk  37  ft.  long  4  ft.  6  in.  wide  with  brick,  @  $1.36 
a  □  yard,  the  bricks  to  be  laid  on  edge. 

4.  A  cellar  with  cement,  @  39^  a  □  yard ;  dimensions,  27  ft. 
X  31  ft. 

5.  Find  which  would  be  the  cheaper :  to  brick  a  sidewalk 
4  ft.  wide  and  275  ft.  long,  @  11^  a  □  foot,  or  to  lay  a  stone 
walk  3  ft.  6  in.  wide  and  of  the  same  length,  @  $1.89  per  □  yard. 


268  STANDARD  ARITHMETIC. 

Bins,  Tanks,  and  Cisterns. 
It  must  be  remembered  that 
2150.42  cubic  inches  —  the  contents  of  a  bushel  measure. 
231  "  "       =    "  "  "    gallon  liquid  measure. 

1.  How  many  bushels  of  wheat  may  be  contained  in  a  box 
measuring  5  ft.  long,  5  ft.  wide,  and  5  ft.  deep  ? 

Note. — If  any  number  of  cubic  feet  be  diminished  by  1/5  of  itself,  the  remainder 
will  represent  very  nearly  an  equivalent  in  bushels,  stricken  measure. 

Thus,  the  box  above  mentioned  contains  125  cubic  feet,  1/5  being  deducted  the 
remainder  is  100,  which  is  the  number  of  bushels  to  within  less  than  1j2  bu.  (.44). 

2.  First  estimate  and  then  compute  exactly  the  number  of 
bushels  of  grain  in  a  box  measuring  3  by  5  by  6  feet ;  also  in  one 
measuring  21/2  by  6  by  7  feet.  What  is  the  difference  between 
the  estimated  and  exact  contents  in  each  case  ? 

3.  What  would  be  the  difference  between  the  estimated  and 
exact  number  of  bushels  in  a  bin  8  by  7  by  5  feet  ? 

4.  I  wish  to  build  a  tank  4  ft.  square  to  contain  700  gallons ; 
how  deep  must  it  be  ? 

5.  How  many  gallons  will  fill  a  circular  reservoir  155  ft.  in 

diameter  and   15   ft.  deep  ?     (The  contents  of  a  circular  reservoir  or  cistern 
is  .7854  of  a  square  one  of  equal  depth  and  having  sides  equal  to  the  diameter.) 

6.  How  many  gallons  of  water  in  a  cistern  9  ft.  in  diameter 
and  10  ft.  deep  ?  

Estimating  the  Weight  of  Hay  in  a  Mow. 

272.  The  average  weight  of  450  cubic  feet  of  meadow  hay, 
or  550  feet  of  clover,  dry  and  well  settled  in  large  mows  or  stacks, 
is  about  one  ton. 

1.  The  average  height  of  the  hay  in  a  mow  is  11  %  ft.,  the 
length  of  the  mow  is  30  ft.,  and  the  width  18  ft.  What  is  the 
estimated  weight  of  the  hay  ?  How  much  heavier  than  clover 
occupying  the  same  space  ? 

2.  The  length,  width,  and  height  of  a  stack  of  clover  average 
12  by  12  by  10  ft.     What  is  its  estimated  weight  ? 


PRACTICAL  MEASUREMENTS.  269 

Miscellaneous. 

1.  The  pumping-engine  at  the  Saratoga  water-works  in  one 
week  pumps  15,307,558  gallons  of  water.  How  long  should  a 
reservoir  150  wide  and  35  ft.  deep  be  to  hold  that  quantity  of 
water  ? 

2.  At  $2.80  per  M,  what  must  be  paid  for  the  shingles  for  a 
barn  having  a  gable  roof  37  ft.  6  in.  long,  and  each  slope  being 

16  ft.  6  in.  wide  ?     (1000  shingles  of  good  quality,  laid  4  in.  to  the  weather, 
cover  120  □  feet.) 

3.  How  many  cords  of  wood  can  be  piled  under  a  shed  45  ft. 
long,  48  ft.  wide,  and  12  ft.  high,  and  what  would  the  wood  be 
worth  at  $4. 75  a  cord  ? 

4.  How  many  gallons  will  fill  a  water-tank  8%  X  6y4  X  5  ? 
How  many  bushels  of  wheat  would  the  tank  contain  ?  How 
many  bushels  of  potatoes  or  apples  ? 

Note. — In  measuring  grains  the  measure  is  stricken,  or  leveled,  but  in  meas- 
uring potatoes,  apples,  etc.,  the  measure  is  heaped.  The  bulk  of  a  bushel  stricken 
measure  is  1/5  less  than  of  the  heaped  measure,  and  the  bulk  of  the  heaped  is  */4 
greater  than  that  of  the  stricken. 

5.  What  will  it  cost  to  slate  a  gable  roof,  each  slope  being  35 
ft.  long  by  18  ft.  wide,  @  $14.75  a  square  ? 

6.  In  a  pile  of  wood  175  ft.  long,  16  ft.  wide,  7  ft.  6  in.  high, 
how  many  wagon  loads  of  cord-wood,  '%  cord  to  a  load,  and  what 
will  it  cost  at  $5.65  a  cord,  50^  a  load  being  paid  for  hauling  ? 

7.  A  surveyor,  in  measuring  a  road,  finds  that  it  is  873  chains 

long.      How  many  miles  is  that  ?     (See  Table,  page  202,  Art.  203.) 

8.  On  £ach  side  of  a  lane  17  chains  long  a  farmer  puts  a  fence. 
How  many  rods  of  fence  does  he  build  ? 

9.  If  a  street  vender  buys  chestnuts  at  $2  per  bushel,  and  sells 
them  at  10^  per  quart,  liquid  measure,  how  much  does  he  gain 
per  bushel  ? 

10.  What  was  the  cost  of  excavating  a  cellar  47  ft.  6  in.  X  39 
ft.  6  in.  and  8  ft.  6  in.  deep,  @  65^  a  cubic  yard  ? 


270  STANDARD  ARITHMETIC. 

Original  Problems. 

Suggestions   to  Pupils. 

1.  Take  measurements  of  the  school-room  for  finding  the  cost 
of  joists,  flooring,  plastering,  etc.  A  comparison  of  the  measure- 
ments taken  will  suggest  corrections  of  errors. 

2.  Take  the  actual  measurement  of  some  rectangular  room, 
give  the  dimensions  with  such  other  information  as  is  necessary 
to  find  the  cost  of  carpeting  or  papering  it. 

3.  Give  the  dimensions  of  a  bin  or  box  to  find  the  quantity  of 
grain,  or  potatoes,  or  apples  it  will  contain. 

4.  Give  the  dimensions  of  a  lot,  and  such  details  of  informa- 
tion as  may  be  needed  to  find  the  cost  of  fencing  it.  An  inspec- 
tion of  a  fence  already  constructed  and  inquiries  made  of  work- 
men will  suggest  the  points  needed. 

5.  Tell  where  some  sidewalk  about  your  school-house  is  needed, 
and  ask  the  members  of  the  class  to  find  what  its  cost  would  be. 
The  pupils  may  determine  what  kind  of  walk  they  would  have, 
and  learn  what  it  would  cost  per  square  yard  or  foot. 

6.  Take  the  measurements  of  a  load  or  pile  of  cord  wood  ;  re- 
port the  same  to  the  class  and  ask  how  many  cords  and  what  it 
would  cost  at  prevailing  prices. 

7.  Find  how  many  cubic  feet  of  coal,  such  as  is  commonly 
used  in  your  neighborhood,  are  estimated  to  weigh  a  ton  ;  report 
the  same  to  the  class  with  the  size  of  some  coal  bin,  and  ask  how 
many  tons  of  coal  may  be  put  into  it. 

8.  Give  the  thickness  of  ice  formed  at  some  place  near  by,  and 
ask  how  many  tons  can  be  taken  from  any  given  space.    (The  weight 

of  1  cubic  foot  of  ice,  at  32°  Farenheit,  is  57.5  pounds.) 

Note. — It  is  not  designed  that  all  the  pupils  shall  prepare  questions  on  all  the 
topics  suggested,  nor  that  they  should  be  restricted  to  them  alone.  No  pupil  should 
present  a  question  which  he  is  not  ready  to  answer  if  required. 


!  i  IcdSiwlssi  ohmerc'h'antsI  MjfMji|  '!!|:| 


CHAPTER   XIV. 

CALCULATIONS   HAVING    REFERENCE  TO 
STANDARD. 


00  AS  A 


Percentage. 

Illustrations. — l.  A  farm  hand,  who  works  "on  shares,"  re- 
ceives from  one  farmer  an  offer  of  7  bushels  of  corn  out  of  every 
16  bushels  raised,  another  offers  him  2  out  of  5,  another  5  out  of 
12.     Which  is  the  best  offer  ? 

Here  it  is  difficult  to  determine  which  is  the  best  offer,  because  the  standards 
of  comparison,  16,  5,  and  12,  are  different. 

Expressing  the  shares  in  the  form  of  common  fractions,  we  have  7/16,  2/5, 
and  5/i2,  and  reducing  these  to  fractions  having  a  common  denominator,  we  obtain 
105/24o>  96/24o>  and  100/240-  Here  24°  IS  tne  common  standard  of  comparison, 
and  the  several  offers  are  respectively  equivalent  to  105,  96,  and  100  out  of  240 
bushels  produced,  whence  we  see  that  the  first  offer  is  the  best. 

But  the  offer  of  V  bushels  out  of  16,  or  7/16  of  a  crop,  is  equivalent  to  the  offer 
of  7/16  of  each  100  bu.,  or  at  the  rate  of  43  3/4  bu.  out  of  100.  Comparing  all 
the  offers  with  this  standard,  we  find  that  they  are  equivalent  to  43  3/4,  40,  and 
41  2/3  bushels  per  hundred,  respectively ;  whence  we  readily  see  how  the  several 
offers  compare  with  each  other. 

2.  In  like  manner  compare  the  value  of  two  iron  ores,  one  of 
which  produces  52  tons  of  metal  from  65  tons  of  ore,  and  the 
other  42  tons  of  metal  from  56  tons  of  ore. 

Suggestion. — What  common  fractional  part  of  each  ore  is  metal  ?  How  many 
tons  of  metal  can  be  produced  from  100  tons  of  each  ore  ? 

Note. — Because  of  its  simplicity  and  convenience,  1 00  has  been  adopted  as  a 
standard  of  comparison  in  almost  every  department  of  business  and  by  all  civilized 
nations ;  hence  we  hear  of  a  boy's  spelling  a  certain  per  cent,  of  the  words  dictated, 
that  is,  at  the  rate  of  so  many  in  a  hundred,  and  in  like  manner  of  the  merchant 
gaining  or  losing  a  certain  per  cent,  of  the  money  he  lays  out  for  his  goods,  of  an 
increasing  per  cent,  of  children  who  are  near-sighted,  etc.,  etc. 


272  STANDARD  ARITHMETIC, 

Definitions. 

273.  Per  Cent,  is  an  abbreviation  of  the  phrase  per  centum, 

and  signifies  by  the  hundred. 

Caution. — The  abbreviation  cent,  in  the  phrase  per  cent,  has  no  reference  to 
the  cent  of  our  decimal  currency. 

274.  A  Rate  per  cent,  is  a  rate  per  hundred. 

275.  The  sign  %  is  annexed  to  the  rate,  and  stands  for  the 

phrase  per  cent. 

Thus,  7  %  is  read  seven  per  cent.,  .07  %  is  read  seven  hundredths  of  one  per 
cent.,  1/3%  is  read  V3  °*  *  Per  cent.,  .00  1/3%  is  read  1/3  of  one  hundredth  of  1 
per  cent. 

276.  Any  per  cent,  of  a  number  is  equivalent  to  so  many 

hundredths  of  it. 

Illustration. — If  100  marks  be  made,  4  in  line  and  25  I     I     I     I 

in  column,  the  following  questions  may  be  asked  :  till 

What  common  fractional  part,  what  decimal  I    I    I    I 

part,  what  per  cent,  of  the  whole  number,  are        and  22  more  Uneg 
in  the  top  line  ?  etc. 

One  mark  is  what  common  fractional  part,  what  decimal  part, 
and  what  per  cent.,  of  25  marks  ? 

Additional  marks  being  made  at  the  foot  of  a  column,  it  may  be  asked  :  What 
per  cent,  is  added  to  the  marks  of  the  column  ?  that  is,  How  many  additional  marks 
would  be  made  in  all  if  1  were  added  to  each  25  in  the  hundred  ? 


What  fractional  part,  and  what  per  cent,  of  all  the  letters  in 
the  italicized  lines  below,  are  contained  in  the  first  word,  in  the 
first  two  ?  etc.     In  each  part  of  eighty-four  f    In  watchful  ?  etc. 

Manuscript,  importance,  regulation,  house-plant,  county  maps, 
blind-mouse,  eighty -four,  be  watchful,  cash  profit,  spring-halt. 

What  per  cent,  of  all  the  letters  in  either  line  are  a's,  b's  ?  etc. 

What  per  cent,  of  the  letters  in  the  word  Oconomowoc  are  o's  ? 
What  percent,  are  c's  ?  etc.  What  per  cent,  of  the  letters  in 
Ohio  are  vowels  ?  Are  consonants  ?  What  #  of  the  numbers 
from  1  to  100  are  primes  ?    From  101  to  200  ?  etc. 


PERCENTAGE.  273 

SLATE    EXERCISES. 

The  pupil  who  desires  to  become  expert  in  computations  of  percentage  should 
be  able  to  convert  per  cents,  into  corresponding  decimal  and  common  fractions 
almost  at  sight. 

Write  the  decimals  and  the  common  fractions  that  are  equiv- 
alent to  the  following  expressions  (all  common  fractions  should 
be  given  in  lowest  terms)  : 

1.  100      200      300  4.  800        80       160 

2.  250       500       750  5.  140       120         50 

3.  40       400       600  6.  150       240       650 

Example. — What  common  fractional  part  of  a  number  is  equiv- 
alent to  12  %  per  cent,  of  it  ? 

101/ 

12  %0  is  equivalent  to  .12%  =  -^ 

12%__25/       _1/ 
100  ~"     /200  ""   /8 

Or,  since  the  numerator,  121/2,  is  an  aliquot  part  of  the  denominator  100,  the 
reduction  can  be  made  directly  by  dividing  both  numerator  and  denominator  by 
12  %>  thus: 

12%.+ 13%      1/ 
100  ~  12%        /8 

In  like  manner  the  following  rates  per  cent,  are  all  reducible  to  simple  common 
fractions. 

What  decimal  and  what  common  fractions  are  equivalent  to 

1.  2%0       18%0       31%0  5.  41%0       58%0       m^i 

2.  37%0      43%0       56%0  6.  83%0       91%0         6%0 

3.  62%0       68%0       81%0  7.  87%0       93%0        6%0 

4.  8%0       16%0       33%0  8.  3%0         13%0       23%0 

What  per  cents,  are  equivalent  to  the  following  fractions  : 

9.  %      %       %  %      %        14.  .04        .40       '    .41%      .14 

10.  %      %       %  %      %        15.  .75        .075        .91%      .12% 

n.  %*     %       %  %      %        16.  .00%    .187%    .83%      .18% 

12-  %      %       %o  Vio     Vn       17.  .0075     .005        .58%      .37% 

13.  %5    %of%  %of%        18.  .625       .00625     .56%      .00% 


274 


STANDARD  ARITHMETIC. 


Applications.  —  Example.  —  l.  I  bought  a 
horse  for  $280  and  sold  it  so  as  to  gain  25$. 
How  much  did  I  gain  ? 

Analysis. — At  1$  (1  to  a  hundred)  I  would  gain 
$2.80,  and  at  25$  I  would  gain  25  times  as  much.  25  x 
$2.80  =  $70  Ans. 

Or,  since  25$  =  .25  or  T/4  of  a  number,  to  obtain 
25$  of  280  we  may  take  1/4,  of  it,  */4  of  $280  =  $70. 

For  exercises,  see  Case  I,  pages  276-279, 

Example.— 2.  If  the  horse  was  bought  for 
$280  and  sold  so  as  to  gain  $70,  what  per  cent. 
was  gained  ? 

Analysis. — At  1$  I  would  gain  $2.80,  and  to  gain 
$70  the  rate  of  gain  would  have  to  be  as  many  times  1  $ 
as  $70  is  times  $2.80,  which  is  25.     25  x  1$  =  25$  Ans. 

Or,  the  gain  $70  is  70/280  or  1/4  of  the  price  paid. 
*/4  of  any  number  =  25$  of  it. 

For  exercises,  see  Case  II,  pages  280,  281. 

Example. — 3.  If  the  horse  was  sold  so  as 
to  gain  $70  at  the  rate  of  25  $,  what  was  the 
price  paid  f 

Analysis.— If  $70  is  25$  of  the  price,  1$  is  l/t8  of 
$70  =  $2.80,  and  100$  or  the  whole  price  =  100  times 
$2.8  =  $280  Ans. 

Or,  if  70  is  25$  or  1/4  of  the  cost,  the  cost  must  be 
4  times  $70  =  $280. 


Written  Work. 
2.80  =  1$ 
25 
1400 
560 
$70.00  =  25$ 


Written  Work. 
2.80)70.00(25 
560 


1400 
1400 


25X1 


Written  Work. 
25)70.0(2.8  = 
50 
200 
200 


25$ 


1* 


100  X  2.8  =  $280 


For  exercises,  see  Case  III,  page  281 ;  and  Case  IV,  page 
282. 

277.  The  three  principal  cases  of  percentage  are  presented  in 
the  foregoing  examples.     They  are  : 

I.  To  find  a  required  per  cent,  of  a  number. 
II.  To  find  what  per  cent,  one  number  is  of  another. 
III.  To  find  a  number  from  a  given  per  cent,  of  it. 

A  fourth  case  is  added,  which  differs  from  the  third  in  nothing  except  that  the 
rate  per  cent,  to  be  operated  with  is  derived  from  the  given  rate  by  adding  it  to  or 
subtracting  it  from  100.     (See  Case  IV.) 


PERCENTAGE.  275 

Definitions. 

278.  The  result  of  taking  any  per  cent,  of  a  number  is  called 
a  Percentage. 

279.  The  number  of  which  a  percentage  is  given  or  on  which 
it  is  to  be  computed,  is  called  the  Base. 

280.  The  base  plus  the  percentage  is  called  the  Amount. 

281.  The  base  minus  the  percentage  is  called  the  Difference. 

Rules. 

I.  To  And  the  Percentage,  the  Base  and  the  Rate  %  being  given. 
Mule.— Multiply  1  %  of  the  base  by  the  rate  %. 

II.  To  find  the  Rate  #,  the  Base  and  Percentage  being  given. 
Mule. — Divide  the  percentage  by  1  %  of  the  base. 

III.  To  And  the  Base,  the  Rate  %  and  the  Percentage  being  given. 
Mule.— Divide  the  percentage  by  the  rate  %  and  multiply  the 

quotient  by  100. 

Formulas. 

282.  The  process  of  finding  a  given  per  cent,  of  a  number 
may  be  indicated  by  signs,  as  follows  : 

I.  Rate  %  x  1%  of  the  Base  =  Percentage. 

Since  the  rate  per  cent,  is  thus  presented  as  the  multiplier, 
1$  of  the  base  as  the  multiplicand,  and  the  percentage  as  the 
product,  we  derive  from  this  formula  two  others,  as  follows  : 
II.  Rate  <f>  =  Percentage  -i-  1  #  of  Base. 
III.  1  £  of  Base  =  Percentage  -f-  Rate  #. 

Note. — It  is  best  that  the  pupil  should  not  become  accustomed  to  depend  on 
the  formulas  as  he  should  not  rely  on  rules  to  direct  his  solutions ;  but  if  he  uses  a 
formula  at  all,  it  is  best  that  he  should  refer  exclusively  to  the  first.  If  he  under- 
stands that,  it  will  suggest  the  others  readily  enough. 

283.  Let  the  learner  accustom  himself  to  compute  percentages  in  the  short* 
est  way  possible.  Thus,  since  25$  is  equivalent  to  .25  or  */4  of  a  number,  we 
may  obtain  25$  of  it  either  by  multiplying  the  number  by  .25  or  by  l/4,  that  is, 
taking  x/4  of  it.     12  72$  is  equivalent  to  .12x/2  or  */•  of  a  number,  etc. 


276 


STAtfDA&D  ARITHMETIC. 


Case  I.— To  find  the  Percentage,  the  Base  and  the  Rate  %  beiny 
given. 

EXERCISES. 
It  is  supposed  that  the  majority  of  pupils  will  be  able  to  note  the  results  in  the 
following  exercises  without  the  aid  of  written  solutions.     As  far  as  practicable,  the 
work  should  be  purely  mental. 


Find 

1.  20#  of  5; 

of  25; 

of  35; 

of  45; 

of  75. 

2.  25#  of  4; 

of  28; 

of  36; 

of  44; 

of  144. 

3.  4#  of  25  ; 

of  75; 

of  125  ; 

of  175 ; 

75$  of 

of  350. 

4.  12%$  of  1600 

;        62%$  of  4000; 

8000. 

6.  16%$  of  1860 

;        33%$  of  2424; 

18%  of  1600. 

6.  9$  of  900; 

7$  of  800 ; 

12$  of  1200. 

7.  8%#of  24; 

of  72; 

of  1440 ; 

of  84; 

of  90. 

8.  37%$  of  40; 

of  84; 

of  4000 ; 

of  96; 

of  14. 

9.  66%$  of  36; 

of  69; 

of  36000 

;      of  53  ; 

of  71. 

10.  6%$  of  64; 

of  32; 

of  64000 

;      of  76  ;  . 

of  80. 

11.  31%$  of  80; 

of  20; 

of  80000 

of  60; 

of  75. 

12.  87%$  of  12; 

of  94; 

of  12000 

;      of  86 ; 

of  63. 

13.  8$  of  37%; 

of  62%; 

of  87%; 

of  6%; 

of  4%. 

14.  9$  of  22%; 

of  33%; 

of  66%; 

of  77%; 

of  44%. 

15.  50$  of  %; 

of%6; 

of8/.; 

of  % ; 

of  "/21. 

16.  6$  of  150; 

of  375  ; 

of  245  ; 

of  180 ; 

of  65. 

17.  16%  0  of  12; 

of  42; 

of  54; 

of  66; 

of  72. 

18.  37%$  of  32; 

of  48; 

of  1.6; 

of  2.4; 

of  5.6. 

Note. — By  taking  any  common  per  cent,  of  two  or  more  bases  separately,  and 
adding  the  percentages  thus  obtained,  the  same  result  is  reached  as  by  taking  a  like 
per  cent,  of  the  sum  of  the  bases. 

Thus,  by  taking  20$  of  5,  25,  35,  45,  and  75,  we  obtain  1,  5,  7,  9,  and  15,  the 
sum  of  which  is  37;  and  by  taking  20$  of  185,  which  is  the  sum  of  the  bases 
5,  25,  35,  45,  and  75,  we  obtain  the  same  result,  37. 

In  this  way  each  pupil  may  test  the  correctness  of  his  results  when  a  common 
per  cent,  of  two  or  more  bases  is  required,  as  in  the  several  lines  of  the  foregoing 
exercises.     No  answers  are  given  to  these  exercises. 


SLATE    EXERCISES. 

&i  i 

19.  Find  41%*  of  $97.68. 

m 

Solution. 

1$  of  $97.68  = 

=  $.9768. 

41 ! 

y3  =  4i%x 

$.9768  =  $40.70. 

Find 

20.  31$  of  6.25 

;       of  0.75; 

of  4.55; 

of  16%; 

of  19%. 

21.  29$  of  752; 

of  8.61; 

of  58.4; 

of  .378; 

Of2%3- 

22.  105$  of  100 

;       of  20  ; 

of  120 ; 

of  140 ; 

of  160. 

23.  117$  of  49; 

of  51  ; 

of  79; 

of  117 ; 

of  %. 

24. 

25. 

26. 

27. 

28. 

Find  20$  of 

16%$  of 

37%$  of 

18$  of 

34$  of 

$7.80 

$18.37% 

$24.85 

$13.12% 

$42.75 

5.30 

15.25 

19.23 

9.56 

83.16 

2.70 

32.18% 

37.82 

12.21 

19.88 

6.20 

17.75 

19.06 

15.66% 

71.21 

1.18 

12.62% 

18.71 

5.18% 

88.45 

2.27 

52.40 

15.01 

1.87% 

.18 

4.38 

64.80 

18.27 

2.25 

.24 

1.07 

51.13 

1.65 

12.68% 

1.31 

Caution. — Let  it  be  remembered  that  20$  of  an 

y  number  =  l{ 

5  of  it,  that 

16  2/3  %  —  x/6,  etc.     Shorten  the  work  as  much  as  possible. 

No  answers  are  given  to  Examples  24-28.  The  correctness  of  results  may  be 
tested  by  comparing  the  sum  of  the  percentages  with  the  like  per  cent,  of  the  sum. 
(See  Note,  page  276.)  Each  column  may  be  conveniently  divided  into  two  or  more 
shorter  ones. 

Loss  and  Gain. 

Example.— l.  A  lot  is  bought  for  $1285 
and  sold  at  a  gain  of  15  $.  How  much  is 
gained,  and  what  is  the  selling  price  ? 

Analysis.— 1%  of  the  cost  is  $12.85,  and  15$  of 
the  cost  is  15  times  $12.85  =  $192.75,  which  is  the 
sum  gained.  The  gain  $192.75  being  added  to  the 
cost  of  the  lot,  we  obtain  the  selling  price,  $1477.75         $1477.75  Selling  price 


$12.85  = 

:1? 

{,  of  cost 

m    15 

6425 

1285 

$192.75  = 

:   15 

* 

1285 

278 


STANDARD  ARITHMETIC. 


2.  If  the  lot  is  bought  for  $1285  and  sold  at  a  loss  of  15  $,  how 
much  is  lost,  and  what  is  the  selling  price  ? 

Here  the  first  step  of  the  solution  is  the  same  as  that  of  the  preceding  problem, 
but  inasmuch  as  there  is  a  loss  of  15$  in  this  case,  $192.75  is  deducted  from  the 
purchase  price  to  find  the  selling  price. 

284.  When  the  purchase  price  and  gain  or  loss  per  cent,  are 
given  to  find  the  selling  price,  the  following  is  the  more  convenient 
process : 

$2  88  =  1  io  3.  If  I  buy  goods  for  $288  and  sell  them 

145  at  a  gain   of  45$,    what  do   I  receive  for 

1440  them  ? 

1152  Analysis. — If  the  goods  are  sold  at  a  gain  of  45$, 

288  they  are  sold  for  145  %  of  what  they  cost  me.     1  %  of  the 

4417  60  ==  145/         cost  *s  $2-88>  and  145  f°  ls  145  times  $2.88,  which  is  equal 

/0  to  $417.60  Ans. 

If  the  goods  were  sold  at  a  loss  of  45  %,  the  selling  price  would  be  55  %  of  what 

they  cost  me.     1  %  of  $288  =  $2.88,  and  55$  =  55  x  | 


4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 
14. 


To     $75   add    33%* 


1.25  " 
.48  " 

1.26  " 
1.54  " 

.81%" 

.72  " 

.84  " 

.72  " 

.96  " 

1.44  " 


37%  i 
62%* 

18* 
25* 
16%* 
45* 
37%* 
12%* 
116%* 


15.  From 

16. 

17. 

18. 

19. 

20. 

21. 

22. 

23. 

24. 

25. 


=  $158.40  Ans. 

.64  take  18%; 

20* 
15* 
45* 
83% 


.18 
2.65 
1.92 

.94 
1.57 
9.85 
6.88 
3.04 
5.75 
9.58 


14* 

23* 

56%* 

43%* 

23%* 

87%* 


26.  Find  12%*,  18%*,  25*,  6%*,  and  37%*  of  $147.36, 

and  add  together  the  percentages.      (Why  is  the  sum  equal  to  the  base  ?) 

27.  If  the  sum  of  18%*  and  31%*  of  1876  be  subtracted 
from  1876,  how  many  will  remain  ?    (Solve  orally.) 

28.  How  many  will  remain  if  2%*  of  897,659  be  subtracted 
from  12  %*  of  the  same  number  ?    (Solve  orally.) 


PERCENTAGE.  279 

Applications. — l.  A  little  boy  who  has  8  apples  gives  25  £  of 
them  to  his  brother,  12  7>$  to  his  sister,  and  50$  to  his  mother. 
What  per  cent,  and  how  many  has  he  left  ? 

2.  Charles  sold  his  sled,  which  had  cost  him  $1.75,  at  20$ 
below  cost.     How  much  did  he  get  for  it  ? 

3.  A  lot  of  damaged  calicoes  are  to  be  sold  at  75  $  below  the 
marked  price.  What  prices  must  be  asked  for  those  that  are 
marked  8^,  10^,  12  %&  160,  20^,  30^  ? 

4.  A  grain  dealer  bought  wheat  for  $9384,  and  sold  it  at  a 
gain  of  472$-     What  did  he  receive  for  it  ? 

5.  If  a  man  owes  $2500,  and  agrees  to  pay  it  in  4  instalments, 
the  first  to  be  50$  of  the  whole,  the  second  25$,  the  third  15$, 
the  fourth  10  $,  what  will  each  instalment  be  ? 

6.  A  man  having  1000  bushels  of  apples,  sold  5  $  of  them  at 
$1.25  per  bushel;  8$  of  the  remainder  at  $1  per  bu. ;  50$  of 
what  was  then  left  at  75^  per  bu.,  and  the  rest  at  60^  per  bu., 
thus  receiving  10$  more  than  he  paid  ;  how  much  did  he  pay  for 
the  whole  quantity  ? 

7.  Mr.  Brooks  bought  a  farm,  which  was  in  very  poor  condi- 
tion, for  $1586  ;  and,  after  two  years  of  careful  cultivation,  which 
paid  for  itself  with  some  improvements,  he  sold  it  for  65$  more 
than  he  paid  for  it.     What  did  he  sell  it  for  ? 

8.  The  number  of  inmates  in  a  workhouse  5  years  ago  was 
110  ;  this  number  has  since  increased  180$.  How  many  inmates 
are  there  now  ? 

9.  A  merchant  bought  goods  for  $297.70,  and  paid  an  addi- 
tional sum  equal  to  7$  of  the  purchase  price  for  cartage,  freight, 
etc.     What  must  he  sell  them  for  to  gain  40$  on  the  whole  cost  ? 

10.  In  a  mixture  of  alcohol  and  water  85$  is  alcohol.  How 
many  gills  of  alcohol  in  3  gallons  of  the  mixture,  and  how  many 
gills  of  water  ? 

11.  560  bushels  of  wheat,  bought  at  $1.10  per  bu.,  were  sold 
at  a  profit  of  10$.     What  did  the  wheat  sell  for  ? 


280 


STANDARD  ARITHMETIC. 


Case  II.— To  find  the  Rate  &  the  Base  and  Percentage  being 
given. 

ORAL    EXERCISES. 

What  per  cent,  of 

1.  10  is  1  ?  5  ?  10  ?  20  ?  30  ?  40  ?  50  ?  60  ?  70  ?  80  ? 

2.  50  is  9  ?  12  ?  15  ?  18  ?  30  ?  45  ?  50  ?  100  ?  125  ? 

3.  200  is  25?    75?   125?   250?   12%?   87%?   16%?   62%? 


SLATE     EXERCISES. 

4.  $212.62%  is  what  per  cent,  of  $486  ? 


Solution. 

$212.62%  -v-  $4.86  =  43.75. 
43.75  X  1$  =  43%#    Ans. 


Analysis.  —  1  %  of  $486  =  $4.86, 
and  $212.62  :/2  is  as  many  per  cent,  as 
$212.62  l/t  are  times  $4.86,  which  is 
433/4.    43  3/4  times  1  %  =  43  3/4  %  Am. 


What  per  cent,  of 
6  *  225  is  9  ? 

6.  %  is  .03  ? 

7.  5%  is  5.5? 

8.  .25  is  .0175  ? 

9.  6.45  is  .32%? 
io.  1  is  %0  ? 

11.  45  is  .3  ? 

12.  .1879  is  18.79? 

13.  55  is  167%? 


11.25  ? 

.045? 

1.1? 

.27? 
.25  %? 

%5? 

.25? 

187  % : 


29.25  ? 
.0.6? 

5.22%? 

.3? 

.451%? 

%o? 
.36? 
281.85? 


33.75? 

.075? 

27.? 
.295? 


38.25? 

.09? 

.825? 

.337%  ? 


.580%?  1.29? 

%?  %? 

.15  ?  .05  ? 

319.43?  394.59? 


2000?        660.27%?     550.22?     112%?   . 

14.  What  per  cent,  is  26%,  29%,  33%,  36%,  of  175? 

15.  What  per  cent,  is  49.5,  56.25,  58.50,  63,  of  225? 

16.  What  per  cent,  is  .024%,  A%   .06%,,  .09%,  of  %  ? 

17.  What  per  cent,  is  .4%,  4.9%,  4.67%,  1.3%,  of  5%? 

*  Answers:  4$,  5%,  13$,  15$,  and  17$.  The  sum  of  thes^  rates  is  54,  and 
54$  of  225  is  121.5,  which  is  equal  to  the  sum  of  the  given  percentages.  In  like 
manner  the  pupil  may  test  for  himself  the  correctness  of  his  answers  to  the  remain- 
ing questions 


PERCENTAGE.  281 

Applications. — 1.  A  boy  buys  an  old  pair  of  skates  for  50^  and 
sells  them  for  25^.  He  then  buys  a  pair  for  25^  which  he  sells 
for  50^.  What  per  cent,  did  he  lose  on  the  first  pair  ?  What  per 
cent,  did  he  gain  on  the  second  ? 

2.  If  a  dealer  buys  a  hat  for  $3,  and  sells  it  for  $4,  what  $ 
does  he  gain  ?  If  he  buys  it  for  $4  and  sells  it  for  $3,  what  per 
cent,  does  he  lose  ? 

3.  One  hundred  pounds  of  beef  were  sold  for  $6,  having  been 
bought  @  4^  a  lb.     What  per  cent,  profit  ? 

4.  A  retail  dealer  in  boots  and  shoes  sold  50  pairs  of  boots 
for  $300.  They  cost  him  $5  a  pair.  What  rate  per  cent,  did  he 
gain  ? 

5.  A  merchant  bought  goods  for  $500.  What  per  cent,  would 
he  gain  by  selling  them  for  $530  ?  For  $525  ?  For  $550  ?  For 
$540  ?    For  $560  ?     For  $575  ?    For  $600  ?    For  $1500  ? 

6.  The  price  of  a  single  ticket  from  Glenwood  to  New  York 
city  is  30^,  but  20  coupon  tickets  can  be  bought  for  $5.  What 
per  cent,  is  saved  by  buying  coupon  tickets  ?  What  per  cent,  is 
lost  by  buying  single  tickets  ? 

7.  10$  of  a  flock  of  sheep  were  killed  by  dogs;  6%$  of  the 
rest  were  lost ;  33V3$  of  the  remaining  number  were  sold,  and  28 
then  remained.     What  was  the  original  number  ? 

8.  At  harvest  time  a  farmer  sold  60  bushels  of  wheat,  which 
was  25$  of  the  quantity  he  seut  to  mill,  and  what  he  sent  to  mill 
was  40$  of  what  he  kept  over  till  the  next  spring.  How  many 
bushels  had  he  at  first  ? 

9.  When  a  merchant  sold  his  goods  for  $261,  he  gained  twice 
as  much  as  he  would  have  lost  had  he  sold  them  for  $207.     What 

was  his  gain  per  cent.  ?     (How  many  times  the  loss  is  the  difference  between 
$261  and  $207?) 

10.  A  grocer  sold  butter  at  12$  profit.  Had  he  sold  it  for  2$ 
more  per  pound,  he  would  have  gained  20$.  What  did  50  pounds 
cost  him  ? 


282  STANDARD  ARITHMETIC. 

Case  III.— To  find  the  Base,  the  Percentage  and  Bate  %  being 
given. 

ORAL    EXERCISES. 

1.  Ten  apples  are  1/2  of  how  many  apples?     50$   of  how 
many  ? 

2.  Eight  bushels  are  %  of  how  many  bushels  ?    16$  of  how 
many  ? 

3.  25  tons  are  %  of  how  many  tons  ?  25  $  of  how  many  ? 

4.  9  is  %  of  what  number  ?  20$  of  what  number  ? 


EXERCISES. 

5.  234  is  56%$  of  what  number  ? 

Analysis.— If  234  is  56 1/i  %  of  any 

oo  u  ion.  number,  1  %  of  the  number  is  such  part 

234  -f-  56  /4  =  4.16  =  1$.  0f  234  as  is  found  by  dividing  234  by 

416  =  100$  or  the  number.         56  y4,  which  is  4.16  and  100$,  or  the 

number  itself  is  416  Ans. 
Or,  since  56  ^^  of  any  number  equals  9/16  of  it,  234  is  9/16  of  the  number 
sought.     If  234  is  9/16  of  the  number,  the  number  itself  is  16  times  1/9  of  234  = 
416  Am. 

Find  the  number  of  which 

6.  3  is  10$  19.  45  is  5$  %  . 

7.  20  is  20$  20.  22  is  %$  32'  %  1S  16  /s* 

8.  18  is  %$  21.  2  is  80$  33.  99.9  is  1.75$ 

9.  56  is  2y3$  22.  100  is  662/3^>  34.  .001  is  8%$ 

10.  75  is  1$  23.  210  is  105$  35.  81  is  9$ 

11.  125  is  95$  24.  65  is  14%$  36.  195  is  200$ 

12.  40  is  62%$  25.  16  is  33%$  37.  95 V*  is  8y3$ 

13.  7  is  12V2$  26.  35  is  41%$  38.  %  is  0.9$ 

14.  11  is  87%$  27.  525  is  25$  39.  2001  is  %  $ 

15.  20  is  33%$  28.  11%  is  %$  40.  6.25  is  37%$ 

16.  14%  is  14%$  29.  232  is  29$  41.  7  is  2% 

17.  19 %  is  62%$  30.  38  is  3%$  42.  999 %  is  100^ 

18.  5  is  20$  31.  12%  is  12% 0  43.  87%  is  50$ 


PERCENT  A  GE.  283 

Case  IV.— To  find  the  Base,  the  Rate  %  and  the  amount  or  dif- 
ference being  giren. 

I.  The  retail  price  of  a  certain  article  is  68^.     How  much  can 
the  retailer  pay  for  it  to  realize  a  gain  of  33  %  0  ? 

aQj  1oqi/  d    f        f  Explanation. — If  when  the  article  sells  for  680 

bSff  —  166  /3ff  Ot  COSt    there  ig  tQ  be  a  gain  of  33  1/3  per  cent  ^  6g^  mugt  be 

133 Y3)     68  133  1/3%  of  the  cost;  hence  this  problem  is  similar 
3              3  to  that  of  Case  III,  which  is  to  find  the  base,  the  per- 
iod   ATTu  centage  and  rate  %  being  given. 
±vv    )&.yj±  0r^  if  when  thc  artic]e  selis  for  680  tuere  is  t0  be 

.51  =  1  #  a  gain  of  33  x/3  #  =  >'/i  of  the  cost,  680  must  be  4/3 

gi   _  cosf  of  the  cost;  hence  the  cost  must  be  3  times  */4  of  68. 

3  times  J/4  of  68  =  510  Am. 

What  number  increased  by 

2.  10  i  of  itself  equals  110  ?  7.  %  %  of  itself  equals  9.06  ? 

3.  75  *  of  itself  equals  $420  ?  8.  %  #  of  itself  equals  $81.72  ? 

4.  62 %#  of  itself  equals  89.37V2?  91/*       „ .     10 

c   oiw    #u-i#  ,     i   qokp  \     9.  ^?<£  of  itself  equals  $90. 342? 

5.  21.5$  of  itself  equals  32.562?  25^  ^ 

6.  83 ys0  of  itself  equals  $87.12  ?     10.  433/4$  of  itself  equals  $1.38? 

II.  I  am  charged  $2.50  for  a  book,  which  the  bookseller  says 
is  33  y3  io  less  than  it  cost  him.     What  was  the  cost  ? 

Explanation.— If  when  the  book  sells  for  $2.50  „„2/  .9  ^n 

there  is  a  loss  of  33  V,  &  the  $2.50  must  be  66  2/3  %  bb  /3/*-oU 

of  the  cost;  hence  this  problem  also  is  similar  to  3 3 

that  of  Case  III.  200      )7.50 

Or,  since  $2.50  is  66  2/3  %  or  2/3  of  the  cost,  AQ7* 1  e/ 

the  cost  must  have  been  3  times  l/s  of  $2.50  — 


$3.75  Am.  3.75  =  100$ 

What  number  diminished  by 

12.  5$  of  itself  equals  $6.65  ?  16.  87y2$  of  itself  equals  10  ? 

13.  5  i  of  itself  equals  19  ?  17.  162/3$  of  itself  equals  955/18  ? 

14.  20  0  of  itself  equals  80  ?  18.  55/8$  of  itself  equals  67.95  ? 

15.  9$  of  itself  equals  9yl0  ?  19.  %$  of  itself  equals  216.38  ? 

Note. — No  special  rule  is  needed  for  Case  IV,  thc  process  of  solution  being  the 
same  as  that  of  Case  III. 


284  STANDARD  ARITHMETIC. 

Applications. — l.  William  buys  a  penknife  for  20^  and  sells  it 
to  James  for  25^,  and  James  sells  it  to  Fred  for  20^.  What  per 
cent,  does  William  gain,  and  what  per  cent,  does  James  lose  ? 

2.  If  the  25  minutes  of  school  time  given  to  recesses  are  8y3$ 
of  the  daily  session,  how  many  hours  in  the  session  ? 

3.  If  a  book  is  marked  to  be  sold  at  25$  above  cost,  but  it  is 
sold  at  20$  below  the  marked  price,  what  was  the  gain  or  loss 
per  cent.  ? 

4.  If  80  pounds  of  coffee  are  exchanged  for  120  pounds  of 
sugar,  what  $  is  the  coffee  worth  per  pound  more  than  the  sugar  ? 

5.  What  per  cent,  do  I  gain  by  selling  an  article  for  $3  for 
which  I  paid  $2.25  ?  What  per  cent,  do  I  lose  by  buying  an 
article  for  $3  and  selling  it  for  $2.25  ? 

6.  A  drover  sold  a  horse  for  $226,  and  thus  gained  25  $.  What 
did  he  pay  for  him  ? 

7.  The  assets  of  a  business  man  are  $135,700,  which  sum  is 
43$  of  his  debts.     What  is  his  indebtedness  ? 

8.  A  fruit  dealer  sold  a  lot  of  oranges  for  $337.50.  which 
allowed  him  a  profit  of  1272$.     What  did  he  pay  for  them  ? 

9.  A  city  lot  was  sold  for  $25,500,  the  gain  on  the  cost  being 
325$.     What  was  the  cost  ? 

10.  A  grocer  sold  300  bushels  of  potatoes  for  $285,  which  was 
162/3$  less  than  he  had  paid  for  them.  How  much  did  they  cost 
him  per  bushel  ? 

11.  A.  sold  goods  at  a  gain  of  18$.  His  profit  was  $29.70. 
How  much  did  he  sell  them  for  ? 

12.  By  selling  a  lot  of  goods  for  $380,  I  gain  3  times  the  per 
cent,  that  would  be  gained  by  selling  them  for  $340.  What  per 
cent,  is  gained  in  the  latter  case  ?    ($380  —  $340  =  2  times  the  gain.) 

13.  In  the  schools  of  a  village  yesterday  there  were  1235  pupils 
present,  which  was  95$  of  the  whole  number  belonging.  How 
many  belonged  to  the  schools  ? 


PERCENTAGE.  285 

Trade  Discount. 

285.  A  discount  is  a  deduction  from  a  price,  from  the 
amount  of  a  bill,  or  other  account. 

286.  In  some  branches  of  business  it  is  customary  to  have  fixed  price  lists 
of  certain  kinds  of  goods,  and,  when  a  rise  or  fall  of  prices  occurs,  instead  of 
changing  every  price  on  a  long  list,  the  rate  of  discount  is  changed. 

287.  The  fixed  price  is  called  the  List  Price,  and  the  dis- 
count is  called  Trade  Discount.  The  Net  Price  is  the  list  price 
minus  the  discount. 

Example. — l.  If  penknives  of  a  certain  quality  are  sold  at  $18 
per  doz.,  with  a  discount  of  33  l/9jfs  what  is  the  net  price  ? 

2.  How  much  must  be  paid  on  a  bill  of  $5560  for  books  if 
20  0  discount  is  allowed  on  account  of  the  great  number  of  books 
sold,  and  a  second  discount  of  5$  is  made  for  cash  ? 

Each  successive  discount  is  made  from  the  results  of  preceding  discounts. 

Find  the  net  prices  : 

List  prices.  Discounts. 

8.  $5.37        250  and  330 

9.  $4.82        400     "    300 

10.  $6.72        300,  100,  and    50 

11.  $3.98        400,  200,    "     100 

12.  $4.97        500,  100,    "     10  0 

Note. — To  find  a  single  direct  rate  of  discount  equivalent  to  two  successive  dis- 
counts, deduct  from  the  sum  of  the  two  rates  either  per  cent,  of  the  other. 
Thus :  GO  and  10  off  =  60  +  10  -  10#  of  60  =  70  -  6  =  64. 


List  prices. 

3.  $5.40 

Discounts. 

250  and  100 

4.  $6.56 

5.  $8.35 

6.  $7.80 

400  "  200 
600  "  50 
500     "    300 

7.  $6.75 

100      "      100 

13.  A  bill  of  hardware  at  list  prices  amounts  to  $276.98,  the 
discounts  are  400,  12*40,  and  100.     What  is  due  on  the  bill  ? 

14.  What  is  the  difference  on  a  bill  of  $780,  between  a  direct 
discount  of  250  and  successive  discounts  of  100,  100,  and  50  ? 

16.  If  the  list  price  of  a  certain  size  and  quality  of  slates  is 
$12  per  gross,  shall   I  gain  or  lose  by  buying  15  gross  of  Mr. 
Brown,  whose  discounts  are  250  and  100,  instead  of  from  Mr. 
Green,  whose  discounts  are  200  and  100  and  50  ? 
13 


286  STANDARD  ARITHMETIC. 

Insurance. 

Insurance  is  security  against  loss  by  fire,  water,  accident,  etc. 

Life  Insurance  is  a  contract  for  the  payment  of  a  specified  sum  at  the  death 
of  the  insured  or  at  the  end  of  a  specified  time,  though  he  may  be  still  living. 

The  Premium  is  the  sum  paid  for  insurance.  It  is  usually  computed  at  a 
given  rate  per  cent,  on  the  sum  insured. 

The  Policy  is  the  written  contract  between  the  insurer  and  the  insured. 

The  insurer  is  called  an  Underwriter,  because  his  name  is  written  under  the 
policy.  

ORAL     EXERCISES. 

1.  What  will  be  the  premium,  if  I  insure  my  house  for  $2000 

at  10?    Aty>0?    At  20?    At%0? 

2.  What  is  the  premium  on  an  insurance  of  1600,  $400,  $800, 
$1900,  $2400,  $100,000,  at  10  ?  At  20  ?  At  y30  ?  At  iy20  ? 
At  iy40?    At  2%0?    At  %0? 

3.  A  vessel  is  insured  for  $45960  at  y4  0.     Find  the  premium. 


WRITTEN     EXERCISES. 

4.  A  match  factory  is  insured  at  4y20;  the  premium  being 
$217.50,  for  how  much  is  it  insured  ? 

5.  A  barn  was  insured  at  the  rate  of  3/40;  the  premium  was 
$19.50.     What  did  the  owner  receive  when  it  was  burned  ? 

6.  At  a  rate  of  5  0,  a  shipper  pays  $213. 95  for  the  insurance 
of  3/4  of  the  value  of  his  goods.     What  was  their  value  ? 

7.  Find  the  the  sum  of  the  premiums  paid  for  the  following 
insurances :  $4000  at  5/80  for  1  year,  $3200  at  iy40  for  2  years, 
$5000  at  iy20  for  3  years,  $2500  at  2y20  for  4  years,  $3500  at 
20  for  3  years,  $2200  at  7/80  for  1  year,  $5400  at  %0  for  1  year, 
$3600  at  2y40  for  5  years,  $4700  at  iy40  for  2  years.     (The  rates 

here  given  are  not  annual,  but  for  the  times  specified.) 

Note. — Great  risks  are  commonly  distributed  in  small  amounts  to  many  different 
companies.     (Why  ?) 

8.  A  building  is  insured  in  19  companies  for  $2500  each,  in 
9  others  for  $5000  each,  and  in  4  others  for  $3500  each.  What 
was  the  total  annual  premium  at  3/5  0  ? 


PERCENTAGE.  287 

9.  The  goods  in  the  building  just  mentioned  were  insured  as 
follows  :  in  1  company  for  $10000,  in  1  for  $9000,  in  16  for  $2500 
each,  in  7  for  $3500  each,  in  4  for  $1500  each,  and  in  1  for  $1000. 
What  was  the  total  premium  paid  annually  at  75^  (per  $100)  ? 

(75^  per  $100  =  .75$  or  3/4#.) 

10.  What  is  the  rate  at  which  a  factory  is  insured  for  $5250, 
if  the  premium  is  $6,5.62%  ? 

11.  The  Ohio  Mutual  Insurance  Company  insured  my  house  for 
$5800  for  a  period  of  3  years  at  1  %  $.     What  was  the  premium  ? 

12.  The  cargo  of  steamer  Gallion,  bound  for  Liverpool,  is 
insured  at  1/2  $.  For  what  sum  is  it  insured,  the  premium  being 
$1500  ? 

13.  My  house  cost  me  $8400.  I  insured  it  for  3/4  of  its  value, 
at  %  $  per  year.  My  books  and  furniture  were  insured  for  $3000 
at  the  same  rate.    What  did  I  pay  annually  for  insurance  on  both  ? 

14.  If  you  have  your  life  insured  for  $5000  at  $15.50  on  $1000 
annually,  what  premium  do  you  pay  ? 

15.  When  30  years  of  age  a  man  insures  his  life  for  $8500,  at 
the  annual  rate  of  $22.70  on  $1000.  If  he  dies  when  60  years 
old,  how  much  more  do  his  heirs  receive  than  he  had  paid  for  in- 
surance ? 

16.  Suppose  the  man  above  mentioned  had  been  insured  from 
his  20th  to  his  60th  birthday,  how  much  would  the  sum  of  his 
annual  premiums  have  fallen  short  of  the  sum  insured,  the  rate 
being  1%0? 

17.  A  manufacturing  company  paid  $214.80  premium  for  in- 
surance on  %  of  the  cost  of  its  buildings  and  machinery,  at  60^ 
per  $100.     What  was  their  cost  ? 

18.  If  in  1  year  an  insurance  company  takes  the  insurance  of 
1000  dwellings  at  2/3i  on  an  average  valuation  of  $3000,  and 
pays  to  its  agents  15  #  of  the  amount  received  for  premiums, 
what  balance  remains  for  profit  and  to  meet  the  expenses  of  the 
company  if  1  of  the  houses  is  totally  destroyed  by  fire  ? 


288  STANDARD  ARITHMETIC. 

Commission  and  Brokerage. 

An  Agent  is  a  person  who  is  authorized  to  transact  business  for  another.  The 
person  for  whom  he  acts  is  his  Principal. 

Commission  is  the  allowance  made  to  an  agent  for  transacting  the  business  of 
another.  It  is  usually  reckoned  at  a  certain  rate  per  cent,  on  the  sum  of  money 
invested  or  realized,  sometimes  at  a  certain  rate  per  bushel,  barrel,  bale,  etc.,  bought 
or  sold. 

Agents  are  known  by  various  names,  as,  commission  merchants,  brokers,  col- 
lectors, correspondents,  etc.,  according  to  the  nature  of  their  business. 

A  Broker  effects  bargains  and  contracts  for  and  generally  in  the  name  of 
others.  The  broker  does  not  take  possession  of  the  property  bought  or  sold.  The 
name  of  broker  is  erroneously  applied  to  those  who  deal  in  stocks,  bonds,  etc.,  on 
their  own  account. 

The  commission  allowed  to  a  broker  is  called  Brokerage. 

A  Commission  Merchant  buys  and  sells  on  account  of  others,  but  in  his  own 
name.  He  has  the  merchandise  in  which  he  deals  within  his  immediate  control.  His 
commission  is  usually  greater  than  that  of  a  broker. 

A  Consignment  is  a  quantity  of  merchandise  sent  by  one  party  to  another. 
The  one  that  sends  it  is  called  the  Consignor;  the  one  to  whom  it  is  sent  is  called 
the  Consignee. 

The  Gross  proceeds  of  a  consignment  is  the  whole  amount  for  which  it  is 
sold ;  the  Net  proceeds  is  the  sum  due  the  consignor  after  deducting  commission 
and  all  other  charges. 

The  calculations  in  commission  and  brokerage  are  simple  applications  of  the 
rules  of  percentage. 

ORAL     EXERCISES. 

1.  If  my  commission  for  selling  an  article  for  $450  is  4$,  how 
much  do  I  receive  ?  How  much  at  4t,%£  ?  At  5$  ?  At  15  #  ? 
At  25$? 

2.  An  agent  sold  a  piano  at  $350,  and  received  $35  commis- 
sion ;  what  rate  per  cent,  is  that  ?  What  rate  if  he  had  received 
$14?    If  $17.50? 

3.  What  will  be  the  fees  of  a  collector  of  taxes  on  $1,200,000 
if  allowed  1%*?    Ifl1/**?    If  1 3/4 ^  ? 

4.  Find  the   commission   on   $200,  $220,  $250,  $300,  $580, 

at  y4&  y8fo. 

5.  A  broker  buys  5  tons  of  currants  at  $8.50  per  cwt.  What 
is  his  brokerage  at  2  fo  ? 


PERCENTAGE.  289 

WRITTEN     EXERCISES. 

6.  An  agent  purchases  5  tons  of  raw  sugar  at  &%#  a  pound, 
and  charges  2yg#  commission.  How  much  money  must  be  sent 
to  him  to  cover  the  cost  and  commission  ? 

7.  I  sell  through  my  broker  7  tons  of  Brazil  nuts  at  $7.50  per 
cwt.    How  much  do  I  receive  if  the  broker  charges  1  fo  for  selling  ? 

8.  A  broker  charged  $74.25  for  effecting  a  loan  of  $3300. 
What  was  the  charge  per  cent.  ? 

9.  A  fruit  broker  sold  $680  worth  of  apples,  and  after  deduct- 
ing 5fo  commission  and  20$  for  freight  and  other  charges,  in- 
vested the  balance  in  oranges.  How  much  did  he  invest  in 
oranges  if  he  charged  2$  for  buying  ? 

Explanation. — Charges  and  commission,  together  amounting  to  25$  of  the 
whole  sum  received  for  the  apples,  being  deducted  from  $680,  there  is  a  remainder 
of  $510,  with  which  the  broker  is  to  buy  oranges  and  pay  himself  2%  on  the  pur- 
chase price. 

If  now  the  broker  were  to  buy  a  dollar's  worth  of  oranges  at  a  time,  and  each 
time  to  pay  himself  2^,  it  is  plain  that  he  would  expend  for  oranges  only  as  many 
times  $1  as  there  are  times  $1.02  in  $510. 

10.  John  Wells  &  Co.  sell  $150  worth  of  eggs  for  W.  Smith, 
charging  him  21/2#  commission.  They  invest  the  proceeds  in 
groceries,  and  charge  2$  for  buying.     How  much  do  they  invest  ? 

n.  A  shoe  manufacturer  forwarded  50  dozen  pairs  of  shoes  to 
his  agent  in  New  York,  who  sold  them  at  $42.60  per  doz.,  charging 
5  io  commission.  He  purchased  leather  with  the  proceeds,  charging 
2fo  for  buying.  What  was  his  total  commission  ?  How  much  did 
he  pay  for  the  leather  ? 

12.  A  cotton  dealer  in  New  Orleans  ships  $10,000  worth  of 
cotton  to  his  broker  in  New  York,  with  instructions  to  purchase 
dry  goods  and  hardware  with  the  proceeds.  The  broker  charges 
2y2$  for  selling  the  cotton  and  2f0  for  buying.  How  much  does 
he  invest,  and  what  is  his  total  commission  ? 

13.  A  commission  merchant  having  sold  a  consignment  for 
$3578,  retains  $95.70  to  pay  charges  amounting  to  $6.25  and  his 
own  commission.     What  rate  per  cent,  commission  did  he  charge  ? 


290  STANDARD  ARITHMETIC. 

14.  A  commission  merchant  sold  500  lb.  of  butter  at  180  per 
pound,  and  invested  the  proceeds  in  oats  at  42^  a  bushel.  He 
charged  4%$  for  selling  and  1%#  for  buying.  What  was  his 
total  commission,  and  how  many  bushels  of  oats  did  he  buy  ? 

15.  Sold  a  consignment  of  merchandise  for  $5000.  What  was 
the  balance  due  the  consignor  after  the  deduction  of  $110.50 
freight,  $250  duty,  cartage,  and  storage,  $75.40  insurance,  and 
5$  brokerage  ? 

Stocks. 

288.  1.  Here  the  pupil  needs  to  know  what  stocks  are,  and  the  meaning  of 
some  of  the  more  common  expressions  used  in  relation  to  them.  To  this  end  the 
following  illustration  will  serve  better  than  mere  definitions. 

2.  Suppose  that  the  citizens  of  a  town  desire  to  have  gas-light  in  their  streets 
and  houses,  and  that  about  $50,000  will  be  needed  to  construct  the  necessary  works. 

3.  No  one  person  or  private  company  will  be  willing  to  risk  so  much  money  in 
the  experiment,  but  if  500  persons  will  take  a  share  in  the  enterprise  and  each 
put  in  $100,  the  plan  can  be  carried  out. 

4.  A  subscription  is  started,  and  it  is  found  that  many  are  willing  to  put  in 
$100,  and  that  some  arc  ready  to  take  two  or  more  shares,  and  possibly  one  or  two 
who  will  take  a  hundred  shares  each.  Thus  it  is  found  that  the  $50,000  can  be 
raised. 

5.  The  subscribers  then  obtain  a  charter,  or  legal  authority  to  act  as  a  company, 
and  appoint  a  Board  of  Directors,  each  subscriber  having  one  vote  in  the  elec- 
tion for  each  share  of  stock  he  has  taken. 

6.  As  soon  as  the  company  is  ready  for  business  the  Board  of  Directors  calls 
for  the  payment  of  the  subscriptions,  either  at  once  or  by  instalments  as  the  money 
is  needed,  until  they  are  all  paid. 

289.  A  certificate  of  stock  is  now  given  to  each  stock  or  shareholder, 
showing  the  number  of  shares  he  has  taken  and  the  price  paid  per  share.  The 
latter  is  called  its  face  or  par  value. 

290.  If  the  company  is  prosperous  and  pays  in  dividends  (division  of 
profits)  more  than  the  money  could  earn  in  other  ways,  the  stock  will  be  at  a 
premium,  that  is,  worth  more  than  its  par  value  ;  but  if  the  dividends  are  small, 
a  share  will  be  worth  less  than  $100 ;  then  it  will  be  said  to  be  below  par,  or  at 
a  discount. 

This  is  an  illustration  on  a  very  small  scale.  The  stocks  of  all  the  incorporated 
companies  in  the  United  States  amount  to  more  than  a  thousand  millions  of  dollars, 
and  there  are  many  men  engaged  in  buying  and  selling  them  in  all  the  great  cities. 


PERCENTAGE.  291 

291.  A  stock  broker  is  one  who  buys  and  sells  stocks  for  others.  For 
buying  or  selling  stocks  in  the  New  York  Stock  Exchange  the  regular  charge  is 
x/8  of  1%  on  their  par  value. 

292*   A  stock.  Jobber  buys  and  sells  stocks  on  his  own  account. 

The  following  report  of  the  number  of  shares  of  certain  stocks  sold  and  the 
highest  prices  paid  for  them  at  the  New  York  Stock  Exchange,  November  16,  1885, 
is  taken  from  a  long  list  to  be  found  in  the  papers  of  the  following  day : 

Sales.  Highest.  Sales.  Highest. 

Adams  Express 10  142  l/t  Central  Iowa 1700  22 1/4 

Atlantic  &  Pacific. ...      850  10  3/s  Central  Pacific 900  47 1/4 

Alb.  &  Susq 15  140  C,  C,  C.  &  1 800  64 l/2 

Atch.,  T.  &  S.  Fe 100  88  Chic,  B.  &  Q 1500  135  lJt 

Canadian  Pacific 1900  55  %  Chicago  &  N.  W.. . .  15,580  115  */4 

Note. — In  the  following  problems  it  is  supposed  that  all  the  transactions  take 
place  through  brokers,  in  behalf  of  outside  parties.  Hence  the  person  for  whom 
stock  is  bought  must  pay  the  price  of  the  stock  plus  the  brokerage,  and  the  seller 
will  receive  the  price  for  which  it  is  sold  minus  the  brokerage. 

Examples. — At  the  above  quotations  : 

1.  How  much  did  the  buyers  pay  for  the  several  stocks  sold 
above  par,  including  brokerage  ? 

Adams  Express. 
Solution. — 10  shares  at  142  l/2  per  share  =  $1425 

Brokerage  at  1/8  £,  1.25 

Cost  of  stock,  $1426.25  Am. 

2.  How  much  did  owners  receive  for  the  several  stocks  sold 
below  par,  brokerage  being  deducted  ? 

3.  What  was  the  brokerage,  at  the  usual  rate,  on  the  purchase 
of  1500  shares  of  Chic,  B.  &  Q.? 

4.  How  much  did  the  buyer  pay  and  how  much  did  seller  re- 
ceive for  1900  shares  of  Canadian  Pacific  ? 

5.  If  I  give  my  broker  orders  to  sell  800  Central  Iowa,  and 
buy  50  Adams  Express,  what  balance  will  he  put  to  my  credit 
after  deducting  brokerage  on  both  sale  and  purchase  ? 

6.  How  many  shares  of  C,  C,  C.  &  I.  could  be  bought  for 
$10,381,  including  brokerage  ?    What  balance  would  remain  ? 


292 


STANDARD  ARITHMETIC. 


Taxes. 

293«  A  tax  is  a  sum  of  money  assessed  on  the  income,  person, 
or  property  of  individuals  for  public  purposes. 

294-.  A  tax  on  property  is  called  a  Property-Tax,  that  on  the  income  is 
called  an  Income-Tax,  that  upon  the  person,  a  Boll-Tax  or  capitation-tax. 

295.  Fixed  property,  such  as  lands,  houses,  etc.,  is  called  Real  Estate. 
Movable  property,  such  as  furniture,  money,  cattle,  merchandise,  etc.,  is  Personal 
Property. 

296.  The  persons  elected  or  appointed  to  estimate  the  value  of  property  to 
be  taxed  are  called  Assessors  or  Appraisers. 

Example. — l.  The  sum  of  money  to  be  raised  by  taxation  in  a 
certain  city  is  8562,600,  the  total  appraised  value  of  the  property 
is  $44,800,000,  and  there  are  25,000  persons  subject  to  a  poll-tax 
of  $1  each.  How  much  will  Mr.  Hunter  have  to  pay,  whose 
property  is  valued  at  $2560  ?     An*.  $31.72. 

Solution.— First  Step. — Subtract  the  $25,000  to  be  received  on  the  polls  from 
the  sura  to  be  levied ;  the  remainder  will  be  the  tax  on  property.  $562,600  — 
$25,000  as  ? 

Second  Step. — Divide  the  tax  on  property  by  the  total  appraised  value  of  the 
property  to  find  the  tax  on  $1.     $537,600  -J-  $44,800,000  =  ? 

The  rate  of  taxation  being  thus  found,  the  tax  on  Mr.  Hunter's  property  is 
readily  ascertained.     To  the  property-tax  must  be  added  his  poll-tax. 

After  the  rate  is  determined,  as  above,  the  computation  of  the  tax  to  be  paid 
by  each  individual  is  greatly  facilitated  by  a  table  like  the  following: 

TABLE  SHOWING  TAXES  AT  THE  RATE  OF  12  MILLS  ON  $t 


$1  pays  $0,012 

$10  pay  $0.12 

$100  pay  $1.20 

$2  pay  $0,024 

$20  ' 

1   $0.24 

$200  "*  $2.40 

$3  "   $0,036 

$30  « 

'   $0.38 

$300  "   $3.60 

$4  "   $0,048 

$40  ' 

'   $0.48 

$400  "   $4.80 

$5  "   $0,060 

$50  ' 

'   $0.60 

$500  "   $6.00 

$6  "   $0,072 

$60  ' 

'   $0.72 

$600  H   $7.20 

$7  "   $0,084 

$70  ' 

'   $0.84 

$700  "   $8.40 

$8  "   $0,096 

$80  ' 

'   $0.96 

$800  "   $9.60 

$9  "   $0,108 

$90  ' 

'   $1.08 

$900  "  $10.80 

The  tax  on  $2,  $3,  etc.,  being  found  by  multiplying  the  rate  by  2,  by  3,  etc.,  the 
rates  for  $10,  $20,  etc.,  are  found  by  removing  the  decimal  points  of  the  first  column 
one  place  to  the  right,  and  for  $100,  $200,  etc.,  two  places,  etc. 


PERCENTAGE.  293 

2.  Find  the  amount  of  taxes  Mr.  A.  has  4nAA.  ._  .  AA 
to  pay  on  property  assessed  at  $2475.  _ 

Explanation. — From  such  a  table  as  the  above  we  -^  0  94. 

would  take  $24.00,  the  tax  on  $2000,  $4.80,  the  tax  on  ~ 

$400,  $.84,  the  tax  on  $70,  and  .06,  the  tax  on  $5,  and  5  =       0.06 

adding  these  together  we  would  find  Mr.  A.'s  tax  on  his  $2475  =  $29. 70 
property  to  be  $29.70. 

3.  My  real  estate  is  estimated  at  $4500,  my  personal  property 
at  $1345,  and  I  have  to  pay  $2  poll-tax.  How  much  tax  will  I 
have  to  pay,  the  rate  being  12%  mills  on  the  dollar  ? 

4.  Find  the  amount  of  taxes  my  neighbor  will  have  to  pay  on 
$9876,  and  $1  poll-tax.     Same  rate. 

5.  Find  the  amount  of  taxes  my  three  neighbors  across  the 
street  pay,  at  the  same  rate,  on  $2732,  $3695,  $8351  ;  each  paying 
$1  poll-tax. 

6.  Find  how  much  a  non-resident  must  pay  on  his  real  estate, 
which  is  listed  at  $6129.     (No  poll-tax.) 

7.  A  person  has  to  pay  $100.20  taxes  at  12  mills  on  the  dollar, 
there  being  no  poll-tax  ?  What  is  the  assessed  value  of  his  prop- 
erty ? 

8.  Suppose  my  property,  real  and  personal,  to  be  listed  at 
$1500,  and  that  I  have  to  pay  3  mills  on  the  dollar  for  state 
purposes,  2  mills  for  county  purposes,  2  mills  for  township  pur- 
poses, 5  mills  extra  for  school  purposes,  and  2%  milts  for  cor- 
poration (village)  expenses ;  how  much  in  all  would  I  have  to  pay  ? 

9.  In  a  state  of  Europe  1  $  is  required  to  be  paid  on  incomes 
from  $100  to  $300,  l'%#  on  incomes  from  $300  to  $500,  and 
2%#  on  incomes  from  $500  to  $800.  Mr.  A.'s  income  was  $450, 
Mr.  B.'s  $175,  Mr.  C.'s  $760.  What  income-tax  did  each  have 
to  pay  ? 

10.  If  my  property  is  valued  at  $2500,  and  the  rate  of  taxa- 
tion for  school  purposes  is  5  mills  on  the  dollar,  what  does  the 
tuition  of  each  one  of  my  three  children  cost  me  if  all  of  them 
attend  the  public  schools  ? 


294  STANDARD  ARITHMETIC. 

11.  Allowing  5/o  for  taxes  uncollectible,  and  2/c  for  collection, 

what  sum  must  be  levied  that  $50,000  may  be  realized  for  the 

building  of  a  school-house  ? 

$1  must  be  collected  for  every  980  needed  for  the  school-house,  because  20  out  of 
every  hundred  go  to  the  collector,  and  $1  must  be  levied  for  every  950  supposed  to  be 
collectible,  since  those  who  do  not  pay  wiil  keep  back  50  of  every  hundred  levied. 

12.  The  people  of  Abdera  wish  to  levy  a  tax  which  will  net 
them  $18,979,  after  paying  the  expense  of  collection,  which  will 
be  3$.  The  assessed  value  of  the  real  and  personal  property  is 
$1,260,000,  and  there  are  323  polls,  each  taxed  $2.  How  much 
will  $1  be  assessed  ? 

13.  Make  out  a  table  similar  to  that  on  page  292. 

From  the  table  (Ex.  13)  find  how  much 

14.  Mr.  W.  M.  Hart  pays  on  $6000        17.  Mr.  H.  Kidd        pays  on  $10000 

15.  Mr.  John  Handy     u     "  $5583        18.  Mr.  L.  B.  Pease     u     "     $7534 

16.  Mr.  E.  G.  Eliot       "      u  $5354        19.  Mr.  R  J.  Luck       u     "     $5821 

20.  For  the  purpose  of  building  a  town-house,  a  tax  of 
$15,961.60  is  to  be  levied  on  property  valued  at  $1,856,000. 
What  will  be  the  tax  on  Mr.  Burns'  property,  which  is  valued 
at  $8650  ? 

21.  A  bridge  costing  $18,135  was  built  by  the  proceeds  of  a 
tax  levied  upon  the  property  of  a  town,  the  rate  of  taxation  being 
50^  on  $100  (5  mills  on  $1),  the  cost  of  collection  being  2yg#. 
What  was4he  assessed  valuation  of  the  property  ? 

22.  If  the  assessed  value  of  the  real  and  personal  property  of  a 
city  is  $80,000,000,  and  a  special  tax  is  desired  for  the  construc- 
tion of  sewers,  what  must  be  the  rate  of  levy  to  realize  $188,160 
for  the  purpose,  if  2$  be  allowed  for  collection  and  4$  of  the 
levy  be  uncollectible  ? 

Note. — The  answers  given  to  problems  such  as  the  preceding  ones  are  based  on 
the  method  of  analysis  given  under  Example  11,  but,  since  the  amount  of  tax  un- 
collectible can  never  be  known  beforehand,  the  sum  to  be  assessed  for  any  given 
use  can  be  determined  with  sufficient  exactness  by  adding  to  the  sum  needed  the 
estimated  percentage  of  taxes  uncollectible  and  the  percentage  charged  for  collec- 
tion. In  States  where  the  collector  is  paid  a  fixed  salary,  the  cost  of  collection 
would  not  be  taken  into  account. 


PERCENTAGE.  295 

Miscellaneous  Problems  in  Percentage. 

1.  Of  480  persons  in  a  village,  30  moved  away  within  one  year. 
What  per  cent,  of  the  whole  number  remained  ? 

2.  If  two  hundred  pounds  of  wheat  make  150  lb.  of  flour,  what 
per  cent,  of  the  weight  of  the  wheat  is  the  weight  of  the  flour  ? 

3.  Twenty  pounds  of  coffee  lose  44/5  lb.  in  weight  by  roast- 
ing ;  what  #  ? 

4.  A  village  of  1253  inhabitants  has  200  children  attending 
school.     What  per  cent,  of  the  whole  population  in  school  ? 

5.  A  person  paid  $22  y2  tax  on  his  income  at  the  rate  of  1%$. 
What  was  his  income  ? 

6.  A  house  was  sold  by  an  agent  for  $5600.  The  agent's  com- 
mission was  iy2#.     How  much  did  the  owner  receive  ? 

7.  A  real  estate  agent  collects  rents  as  follows  : 

For  Mr.  Williams,  $2384.20        John  Jones,    $936.18        Mr.  Cook,  $786.15 
"    Mr.  Johnson,       856.75        Henry  .Jones,  1852.00        Mr.  Doan,    385. 

What  is  the  amount  of  his  commissions  at  3  per  cent.? 

8.  One  and  a  quarter  per  cent,  of  the  inhabitants  of  the  king- 
dom of  Prussia  are  annually  called  into  military  service.  How 
many  men  do  the  city  of  Breslau,  with  240,000  inhabitants,  and 
the  city  of  Hildesheim,  with  23,000  inhabitants,  have  to  furnish  ? 

9.  A  farmer  bought  a  team  of  horses,  but  could  pay  only  $155 
in  cash,  37 yg#  remaining  unpaid.  What  was  the  price  of  the 
horses  ? 

10.  An  inspector  of  coal  mines,  having  a  salary  of  $2400  a 
year,  pays  $560  rent  for  house  and  barn.  iy2#  taxes  on  an  assess- 
ment of  $480,  and  y2$  for  insurance  of  books  and  furniture 
valued  at  $1250.  What  $  of  Ijis  salary  does  he  pay  for  rent, 
taxes,  and  insurance,  respectively  ? 

11.  On  one  occasion  the  price  of  a  barrel  of  petroleum  fell  from 
90^  to  78^.  What  per  cent,  was  the  decline  ?  Shortly  after  the 
price  rose  again  to  90^.     What  per  cent,  was  the  advance  ? 


296  STANDARD  ARITHMETIC. 

12.  Twenty  pounds  flax,  when  spun,  make  17%  lb.  of  yarn. 
What  per  cent.  ? 

13.  Eight  pounds  of  beef  are  reduced  1  lb.  in  weight  by  boil- 
ing and  1%  lb.  by  roasting.  What  per  cent,  of  weight  is  lost  by 
each  process  ? 

14.  If  a  single  railway  fare  to  the  city  is  30^,  what  per  cent, 
would  I  save  by  the  purchase  of  100  tickets  for  $20  ? 

15.  The  Union  Steel  Screw  Co.  declared  a  dividend  of  17%  $ 
upon  its  stock.  What  did  stockholders  receive  who  had  respect- 
ively $900,  $2000,  $4700,  $2300,  and  $1100  worth  of  stock  ? 

16.  A  merchant  sends  out  bills  for  collection  as  follows  : 

$184.75  $57.61  $384.21  $728.13 

136.54  98.13  17.86  564.21 

19.81  156.22  918.54  .    1986.54 

5.78  7.61  12.32  .95 

846.00  387.60  50.65  18.70 

The  collector  receives  6$  on  all  sums  less  than  $100,  4$  on 
amounts  from  $100  to  $500,  and  2$  on  all  sums  greater  than 
$500.     What  will  be  his  commission  if  all  are  collected  ? 

17.  Church  bells  commonly  contain  80$  of  copper,  5.6$  of 
zinc,  10.1$  of  tin,  and  the  rest  is  lead.  At  that  rate,  how  much 
of  each  is  contained  in  the  great  bell  at  Moscow,  which  weighs 
443,772  pounds  ?    What  per  cent,  is  lead  ? 

18.  A  carpet  dealer  reduced  the  price  of  certain  goods  12y2$, 
which  amounted  to  12^  on  the  yard.  What  did  the  goods  sell  at 
per  yard  before  and  after  the  reduction  ? 

19.  Twelve  quarts  of  good  milk  will  give  iyB  qt.  of  cream. 
What  per  cent,  of  the  milk  is  cream  ? 

20.  The  liabilities  of  a  bankrupt  merchant  are  $7200,  his 
assets  only  $3200.  How  much  will  his  creditors  get,  to  whom  he 
owes  respectively  $2572,  $856,  $782,  $1025,  $1912,  and  $53  ? 

21.  The  daily  wages  of  a  workman  were  increased  25^,  or 
6%$.     What  did  he  get  before  and  after  the  increase  ? 


PERCENTAGE.  297 

22.  Charles  has  a  salary  of  $750,  his  brother  $850.  What  $ 
does  his  brother  receive  more  than  he  ?  By  what  per  cent,  is 
Charles's  salary  less  than  his  brother's  ? 

23.  If  eight  pounds  of  imperial  tea  may  be  had  for  $9,  and  a 
single  lb.  of  the  same  kind  costs  $1.20,  what  is  the  per  cent, 
saved  by  buying  8  lb.  at  a  time  ? 

24.  A  gentleman  is  insured  for  $5000.  His  premium  is  $96.25. 
How  much  does  he  pay  on  a  thousand  ?     What  per  cent.  ? 

25.  The  butcher  estimates  a  beef  to  weigh  980  pounds,  of 
which  57$  is  salable  as  meat  and  6y4$  tallow.  What  is  the 
weight  of  the  meat  and  tallow  together  ? 

26.  Colonel  A.  has  to  pay  $1200  rent,  Major  B.  $1000.  The 
owner  of  the  two  houses  raises  the  rent  $100  on  each.  What 
per  cent,  is  the  major's  rent  raised  more  than  that  of  the  colonel  ? 

27.  The  weight  of  a  cubic  foot  of 

American  pine,  when  green,  is  44.75,  when  seasoned,  30.7. 
Ash  u         *       "  58.18       u  "  50. 

Beech  "         "       "  60  u  "         53.37. 

Cedar  "  "       "  32  "  "  28.25. 

English  oak  "         "      u  71.6        "  "         43.5. 

What  per  cent,  does  each  lose  in  weight  by  being  seasoned  ? 

28.  What  per  cent,  income  does  Mr.  Abel  have  more  than 
Mr.  Bain,  if  A.  has  $25X)0  and  B.  $2000  ?  If  A.  has  $3000  and 
B.  $2500  ?  If  A.  has  $2000  and  B.  $1500  ?  If  A.  has  $1750  and 
B.  $1250  ?    If  A.  has  $600  and  B.  $100  ? 

29.  Find  what  per  cent,  the  lower  income  is  of  the  higher  in 
each  of  the  above-mentioned  cases. 

30.  Anna  bought  8  yd.  of  tape  for  5<f  ;  Emma,  25  yd.  for  15^. 
What  per  cent,  did  Anna  pay  per  yard  more  than  Emma  ?  What 
per  cent,  did  Emma  pay  per  yard  less  than  Anna  ? 

31.  Five  men  in  a  factory  accomplish  as  much  work  as  8  boys. 
What  per  cent,  of  a  man's  work  does  a  boy  do  ?  What  per  cent, 
of  a  boy's  work  does  a  man  accomplish  ? 


298  STANDARD  ARITHMETIC, 

32.  If  beech  timber  is  worth  16  %#  more  than  pine,  what  is 
the  value  of  5  cords  of  the  former  when  3  cords  of  the  latter  is 
worth  $12  ? 

33.  In  1885  there  were  90,920,707  shares  of  stock  bought  and 
sold  in  the  New  York  Stock  Exchange.  What  did  the  brokerage 
amount  to  at  %  #  on  both  sale  and  purchase  ? 

34.  In  1885  the  highest  price  paid  per  share  for  Manhattan 
Consolidated  stock  was  123 y2,  the  lowest,  65  ;  for  Louisville  and 
Nashville,  highest,  51%,  lowest,  22;  for  Pacific  Mail,  highest, 
70,  lowest,  463/4.  What  per  cent,  would  have  been  lost  by  buying 
at  the  highest  and  selling  at  the  lowest  rate  in  each  case  ? 

35.  If  a  boy  buys  5  tops  and  sells  4  for  as  much  as  the  5  cost 
him,  what  per  cent,  does  he  gain  on  the  tops  sold  ?  What  per 
cent,  would  he  gain  on  the  tops  sold  if  he  sold  3  for  what  4  cost  ? 
If  he  sold  2  for  what  3  cost  ?     If  he  sold  1  for  what  £  cost  ? 

36.  What  per  cent,  would  the  boy  lose  if  he  sold  the  5  for 
what  4  cost  ?  What  per  cent,  would  he  lose  if  he  sold  5  for  what 
3  cost  ?    5  for  what  2  cost  ?    5  for  what  1  cost  ? 

37.  In  1885  there  were  12,480,423  shares  of  C,  M.  and  St.  P. 
reported  to  have  been  bought  and  sold  in  the  New  York  Stock 
Exchange.  What  did  the  commissions  of  the  brokers  amount  to 
at  y8#  commission  for  buying  and  selling  ? 

38.  If  18  horses  draw  as  much  as  30  oxen,  what  per  cent,  less 
than  a  horse  does  an  ox  draw  ?  What  per  cent,  does  a  horse  draw 
more  than  an  ox  ? 

39.  In  the  schools  of  a  city  there  are  in  the  first  year's  course 
8666  pupils ;  in  the  second,  3205  ;  in  the  third,  3960 ;  in  the 
fourth,  2456;  in  the  fifth,  2012;  in  the  sixth,  1125;  in  the 
seventh,  654 ;  in  the  eighth,  640.  What  per  cent,  of  the  whole 
number  in  each  ?     (What  per  cent,  in  all  ?) 

40.  A  merchant  had  marked  some  calicoes  36^  per  yard,  which 
was  20$  above  cost,  but  finally  sells  them  at  33y3#  below  the 
marked  price.     What  per  cent,  on  first  cost  does  he  lose  ? 


PERCENTAGE.  299 

41.  Mr.  Williams  has  an  insurance  on  his  life  for  $6000.  His 
annual  premium  is  $185,  but  each  year  he  has  the  benefit  of  a 
dividend  of  40$  on  the  premium  of  the  preceding  year.  What 
is  the  net  cost  of  his  insurance  per  year  9    (40  #  of  $185  =  ?    $185 

minus  dividend  =  ?) 

42.  In  consequence  of  a  rise  in  the  market,  a  merchant  marks 
up  calicoes  10$,  which  were  already  marked  to  sell  at  25$  ad- 
vance on  cost.  What  per  cent,  advance  on  first  cost  is  the  latter 
price  ? 

43.  28,000  bricks  are  needed  for  a  building.  How  many  have 
to  be  ordered,  if  6  %  $  be  allowed  for  waste  ? 

Selling  price.         Gain  per  cent. 

44.  $155.00  3y3$ 

45.  $110.00  25$ 

46.  $23400  4$ 

47.  $95.00         '    66%$ 

48.  $187.00  10$ 

49.  $5.60  12$ 
What  was  the  cost  ?  What  was  the  cost  ? 

56.  A  merchant  sold  y4  of  a  certain  lot  of  goods  at  10$  profit, 
V3  at  20$  profit,  and  %  at  15$  profit.  The  remainder,  on  which 
he  lost  5  $,  he  sold  for  $142  %  How  much  did  he  get  for  the 
whole  ? 

What  per  cent,  is  gained  or  lost  when  I  buy 

57.  For     $5  and  sell  for    $7  ?       61.  For   20^  and  sell  for  23^  ? 

58.  "     $30    "  "      $45?       62.     "       4^     "         "         3^? 

59.  "     $25    "  "      $21?       63.     "     $35     "         "     $42? 

60.  "  $40  "  "  $36?  64.  "  $15  *  *  $13%? 
65.  Mr.  James  sent  his  check  for  $2500  to  a  broker,  with  in- 
structions to  buy  good  stocks.  The  broker  bought  bank  stock, 
then  selling  at  105  %  per  share  (par  value,  100).  How  many 
shares  did  he  buy  ?  What  sum  remained  to  Mr.  James's  credit 
after  deductinp;  the  broker's  commission  ? 


Selling  price. 

Logs  per  cent. 

50. 

$6.80 

25^ 

61. 

$117.00 

16%  * 

52. 

$25.50 

3%* 

63. 

$30.00 

66  %* 

64. 

$3.20 

87^ 

55. 

$72.00 

8%* 

300  STANDARD  ARITHMETIC. 

Original  Problems. 

1.  Construct  five  problems,  each  pertaining  to  business  trans- 
actions : 

1.  Giving  a  base  and  rate  per  cent.,  to  find  the  percentage. 

2.  Giving  base  and  percentage,  to  find  the  rate. 

3.  Giving  rate  and  percentage,  to  find  the  base. 

4.  Giving  a  rate  and  amount,  to  find  the  base. 

5.  Giving  rate  and  difference,  to  find  the  base. 

2.  Get  from  some  friend  engaged  in  any  line  of  business  a 
statement  of  some  transaction  requiring  a  calculation  in  percent- 
age, and  form  a  proper  question  for  the  class.  This  may  be  in 
the  line  of  trade  discount,  insurance,  taxes,  etc.,  etc. 

3.  Find  in  the  daily  papers  statements  of  stock  sales.  They 
will  furnish  a  great  variety  of  problems. 

4.  On  occasion  of  a  fire  in  your  city  or  neighborhood,  ascer- 
tain the  facts  concerning  insurance,  and  inquire  what  advantage 
was  gained  by  insurance,  if  any,  or  what  loss  resulted  from  fail- 
ure to  insure. 

5.  Construct  a  problem  in  taxation  on  the  model  of  any  one 
or  more  of  those  found  on  page  294. 

6.  Suppose  yourself  a  commission  merchant  buying  or  making 
sales,  or  both,  for  some  party  in  a  distant  city.  Your  supposition 
may  range  from  a  transaction  in  a  few  bushels  of  hickory  nuts  to 
millions  of  bushels  of  grain. 

7.  Suppose  the  case  of  a  broker  buying  and  selling  real  estate 
for  yourself,  and  ask  his  commission  on  various  imaginary  trans- 
actions. 

8.  Find,  if  you  can,  a  list  price  of  books,  magazines,  or  other 
articles  of  merchandise,  with  the  discounts  given  on  the  same, 
and  ask  the  rates  per  cent,  gained  by  dealers. 

9.  Tell  the  class  the  number  of  different  boys  absent  from 
school  during  any  week  or  month,  and  ask  what  per  cent,  of  the 
whole  number  of  scholars  were  absentees. 


CHAPTER    XV. 


INTEREST. 

297.  Interest  is  compensation  for  the  use  of  money. 

We  pay  rent  for  the  use  of  a  house,  hire  for  the  use  of  a  horse,  interest  for  the 
use  of  money. 

298.  The  Principal  is  the  sum  of  money,  for  the  use  of 
which  interest  is  paid. 

299.  The  Rate  of  Interest  is  the  rate  per  cent,  allowed  for 
the  use  of  money  for  one  year  or  other  specified  time. 

Note. — Any  given  rate  is  understood  to  be  the  rate  for  one  year,  unless  the 
time  be  specified ;  as,  per  month,  per  day,  etc. 

In  this  book,  when  no  rate  is  mentioned,  6  %  is  understood.  Thus,  in  the  ques- 
tion "What  is  the  interest  of  $50  for  6  months?  "  6%  per  annum  is  understood. 

300.  Legal  Interest  is  a  rate  fixed  by  law  for  cases  in  which 
no  rate  is  specified  in  the  agreement  between  the  parties  inter- 
ested. 

In  most  of  the  States  a  limit  is  fixed  to  the  rate  of  interest  which  may  be 
received.  Interest  at  a  rate  exceeding:  the  limit  allowed  by  law  is  termed  usury, 
to  which  some  penalty  is  usually  attached. 

301.  The  Amount  is  the  sum  of  the  principal  and  interest. 

If  we  hire  money  we  return  it  and  pay  interest  for  the  use  of  it,  as  we  return 
a  hired  horse  and  pay  for  his  use. 

302.  Simple  Interest  is  interest  on  the  principal  alone,  and 
is  payable  with  the  principal. 

303.  In  the  common  method  of  computing  interest,  12  months 

of  30  days  each  or  360  days  are  reckoned  as  1  year. 

Note. — Though  this  method  of  reckoning  time  is  not  exact,  it  is  the  most  com- 
mon because  it  is  the  most  convenient,  and  in  most  States  it  is  legalized  by  statute. 


302  STANDARD  ARITHMETIC. 

ORAL      EXERCISES. 

1.  Find  the  interest  of  $600  at  5  $  for  1  year. 

Analysis. — The  interest  of  $600  for  1  year  at  1  %  is  $6,  at  5  %  it  is  5  times 

$6,  or  $30. 

2.  What  must  be  paid  for  the  use  of  $800  for  l1/,  years  at  6$  ? 

Analysis. — The  interest  of  $800  for  1  year  at  1%  is  $8.  For  1  1/2  years  it  is 
1 J/2  times  8  =  $12,  and  at  6%  it  is  6  times  $12  =  $72. 

3.  Find  the  interest  of  $200  for  1  year  at  4^  ;  of  $500  at 
3y80  ;  of  $800  at  Qfc ;  of  $700  at  S/0 ;  of  $1000  at  100 ;  of  $400 
at  5fc. 

4.  Find  the  interest  of  $400  for  2  years  at  d{0:,  of  $500 
at  4y8#;  of  $600  at  ?%£ 

5.  Find  the  interest  for  6  months  of  $1000  at  5  $  per  annum  ; 
of  $150  at  4y8#;  of  $275  at  4^;  of  $1000  at  4%& 


SLATEEXERCISES. 

1.  What  is  the  interest  of  $1230  for  1  year  ?    (What  rate  is  here 

understood  ?) 

2.  Find  the  interest  of  $120  at  &0 ;  of  $140  at  3y80;  of 
$260  at  5  i  ;  for  3  years. 

3.  What  is  the  interest  of  $160  at  6*/4#  for  3  years  ? 

Analysis.— The  interest  of  $160  at  1  %  for  1  year  is  $1.60,  at  6  */4  %  it  is  6  x/4 
times  $1.60,  or  $10,  and  for  3  years  it  is  3  times  $10,  or  $30. 

Find  the  interest  for  one  year  of 

4.  $450  at  4y8#  9.  $2630  at  4y8#  14.  $7428  at  5y8# 

5.  $680  at  3y8#  10.  $4920  at  5$  15.  $9654  at  6$ 

6.  $960  at  rVfjf  11-  $5000  at  33/4$  16.  $7851  at  6y8# 

7.  $840  at  5y8#  12.  $3720  at  ff/|£  17.  $9643  at  7^ 

8.  $1720  at  6y80  13.  $4680  at  4%J*  18.  $5430  at  5^ 

19.  How  much  must  be  paid  for  the  use  of  $80  for  9  months 

at  5^? 

Analysis.— The  interest  of  $80  for  1  year  at  1  %  is  800,  at  5$  it  is  5  times 
80  0  =  $4,  and  for  9  months  it  is  3/4  of  $4  =  $3- 


INTEREST.  303 

Find  the  interest  of 

20.  $72  at  5$  for  4  years.  24.  $56  at  Hi  for  5?/4  years. 

21.  $70  at  4ys#  for  5  years.  25.  $675  at  3y3#  for  6  months. 

22.  $84  at  4y6#  for  2y2  years.  26.  $780  at  5$  for  5  months. 

23.  $97  at  6y4#  for  3y3  years.  27.  $825  at  4#  for  7  months. 

28.  Find  the  interest  of  $228.50  for  5  mo.  18  d. 


$2,285 
6 

Solution. 
Int.  for  1  jr,  at  1 

$13,710 
168 

u       u        u        a  q 

360)2303.280 

- 

$6,398       "      "    5  mo.  18  d.  =  168  d. 

Since  360  d.  are  reckoned  as  1  year,  the  interest  for  168  d.  is 
1C8/360  of  the  interest  of  1  year.  Hence,  we  multiply  $13.71,  the 
interest  for  one  year,  by  168,  and  divide  the  product  by  360,  as 
above. 

Cancellation. — If  we  desired  to  shorten  the  work  by  cancella- 
tion, we  would  arrange  all  the  factors  of  the  dividend  on  the  right 
side  of  a  vertical  line,  and  place  the  divisor  on  the  left,  and  cancel 
as  follows : 

$ttU    $.457 

m     14 

14  X  $.457  =  $6,398  Ans. 

304.  The  computation  of  interest  as  already  presented  does 
not  involve  any  process  or  principle  with  which  the  pupil  is  not 
entirely  familiar.  They  are  such  as  would  be  adopted  by  any  one 
acquainted  with  the  elements  of  arithmetic,  though  he  might  not 
have  had  any  instruction  in  "Interest"  as  taught  in  the  books. 
It  is  therefore  sometimes  called  the  general  method.  But  any 
one  who  wishes  to  become  expert  in  computing  interest  without 
the  aid  of  Tables  should  be  able  to  use  shorter  methods. 


804  STANDARD  ARITHMETIC. 

The  60  Day  Method. 

305.  At  the  rate  of  6^  per  year,  the  rate  for  2  months  or  60 
days  is  1$,  which  is  equal  to  .01  of  the  principal ;  and  for  6  days 
it  is  yl0  of  1$,  which  is  equal  to  .001  of  the  principal. 

Example.— l.  What  is  the  interest  of  $562  for  2  mo.   6  d.? 

Solution. 
Int.  of  $562  for  2  mo.  ==  $5.62 

"       "    6  d.     =      .562 
2  mo.  6  d $6.18 

2.  Find  the  interest  of  $328,  $532,  $690,  $1085,  $52,  $780, 
and  $630  each,  for  2  mo.   6  d.  at  6$  per  annum. 

306.  Thus  we  see  that  when  the  rate  of  interest  is  6$  per 
year  we  may  find  the  interest  of  any  principal  for  two  months  by 
removing  the  decimal  point  two  places  to  the  left,  and  for  six 
days  by  removing  it  three  places  to  the  left,  prefixing  ciphers 
when  needed. 

307.  By  taking  such  multiples  and  parts  of  these  results  as 
the  given  time  requires,  and  adding  them  together,  the  interest 
may  be  found  for  any  given  time. 

3.  What  is  the  interest  of  $280  for  9  mo.   15  d.  ? 

Solution. 
2  mo.'s  int.,  $2.80  ;  6  d.'s  int.,  $.280. 
$8.40  Int.  for    6  mo.  (3  x  2  mo.) 
4.20     "      "      3mo.  0/8of  fimo.) 
.56     "      "    12  d.     (2x6  d.) 
■14     "      "      3  d.     (i/g  of  6  d.) 
$13.30     "      "      9  mo.  15  d.  Am. 

4.  WThat  is  the  interest  of  $3275  for  63  d.  ? 

Solution. 
$32.75       Int.  for  60  d. 
1.6375     "      "      3_<L 
$34.3875     "      "    63~dT  Ans. 

5.  Find  the  interest  of  $48,225  for  93  d. 


INTEREST. 


305 


6.  Find  the  interest  of  $72.85  for  3  years  5  mo.  27  d. 

Complete  Solution. 
.  7285         Int.  for    2  mo. 


14 

570 

Int.  for  40  mo.  (20  x  2  mo.) 

36425 

"      "      1  mo. 

29140 

"      "    24  d.  (4  x  6  d.) 

036425 

"      "     3  d.  C/2  of  6  d.) 

Shorter  Process. 

729 

Int.    2  mo. 

14 

58 

"    40  mo. 

365 

"      1  mo. 

291 

"    24  d. 

036 

"      3d. 

$15. 

272 

"      3  yr.  5  mo.  27  d 

$15  .  262075       "      *      3  yr.  5  mo.  27  it. 

For  business  purpose  it  is  sufficiently  exact  to  carry  the  work 
to  mills  or  to  tenths  of  mills  at  furthest,  as  in  the  shorter  process. 

In  this  process,  when  the  decimal  in 
the  fourth  place  is  less  than  5  it  is  re- 
jected. When  5  or  greater  than  5  the 
number  of  mills  represented  in  the  third 
order  is  increased  by  1 . 

Note.  —  The  result  of  the  shorter 
process  is  1  cent  greater  than  that  of 
the  complete  solution,  but  in  a  business 
transaction  this  difference  would  be  dis- 
regarded.    Answers  are  given  as  found  by  the  shorter  process. 

7.  How  much  must  be  paid  for  the  use  of  $125.25  for  117  days 
at  6$  per  annum  ?    What  will  be  the  amount  ? 
Complete  Solution. 
Principal,  $125.25  ;  Int.  60  d.,  1.2525  ;  Int.  6  d.,  .12525.         Shorter  Process. 

1.253 
627 
418 
125 
021 


1 

2525 

Int. 

60  d. 

62625 

u 

30  d. 

4175 

u 

20  d. 

12525 

a 

6d. 

020875 
442375 

14 

1  d. 

$2 

Int. 

117  d. 

Principal,  $125 

25 

Amount,     $127 

.  692375 

2.444 
$125.25 
$127,694 


8.  Find  the  interest  of  $9280  for  1  yr.  7  mo.  7  days. 

9.  What  is  the  interest  of  $13985  for  2  years  23  days  ? 

10.  What  is  the  interest  of  $18.56  for  1  year  8  mo.  16  days  ? 

11.  What  is  the  interest  of  $198  for  2  years  11  mo.  ? 


306 


STANDARD  ARITHMETIC. 


If  it  be  required  to  find  only  the  amount  of  a  given  sum  at  interest  for  a  given 
time,  the  process  may  be  made  somewhat  shorter  than  in  the  preceding  example 
by  adding  the  principal  with  the  several  items  of  interest,  as  follows : 

12.  Find  the  amount  of  8328  for  2  yr.   11  mo.  26  d. 

Principal,  $328;  Int.  for  2  mo.,  $3.28;  Int.  6  d.,  .328. 


$3.28] 
17 


$55.7( 


28. 

Principal. 

55.76 

Int. 

for  34  mo 

1.64 

a 

"      lmo 

1.312 

M 

"    24  d. 

.109 

a 

"      2d. 

$386,821  Amount. 


Interest  at  other  Rates  than  6°/0. 

the  interest  at  any 


From  the  interest  of  any  given  sum  at  6 
other  rate  can  be  readily  found. 

Example.— What  is  the  interest  of  $329.75  for  1  yr 

23  d.  at  Q><?<>  ?    At  3f0  ?  4tf0  ?   5f0  ?   7^  ?   Sfo  ?   9f0  ? 

Interest  at  6</0. 

Solut'on. 


5  mo. 


Prin.,  $329.75  ;  Int.  2  mo.  =  $3.2975  ;  Int.  6  d.  =  .32975 

Sh 

jrter  Process. 

$26.3800     Int.  1  yr.  4  mo.  (8x2  mo.) 

$26.38 

1.64875     "     1  mo.  (*/,  of  2  mo.) 

1.649 

1.09917     "     20  d.  (*/i  of  60  d.) 

1.099 

.16488     "       3  d.  (V2  of  6  d.) 

.165 

Ans.  $29.29280  Int.  for  1  yr.  5  mo.  23  d. 

Ans. 

$29.-93 

Interest  at  other  per  cents. 

Having  thus  obtained  the  interest  of  $329. 75  for  1  yr.  5  mot 
23  d.  at  6  <fc,  the  interest  of  the  same  sum  for  the  same  time — 


At  80  =  Vi 

of  29.293 

=  $14,647  1 

Or, 

[S%  divide  by  2 

"  H  =  3/3 

a 

u 

=  $19,528 

from 

4$  subtract  1/3 

"  5^=5/6 

a 

U 

=  $24,411 

int.  at 

Z%      "      v« 

"  7#=V. 

it 

ft 

=  $34,175 

Qfc  to 

1%  add  Ve 

a  H  =  *U 

u 

u 

=  $39,056 

find  int. 

8£     "     Va 

"  o*=»/. 

a 

M 

=  $43,941  J 

at 

1:9*     "     V. 

INTEREST. 

307 

SLATE 

EXERCISES. 

Find  the 

interest  : 

Find  the  interest: 

Principal. 

Rate.         Time. 

Principal. 

Rate.              Time. 

1.  $100, 

5  $,    2  yr. 

19.  $840, 

11$,  15  mo.  20  d. 

2.  $100, 

6$,    2  yr.  6  mo. 

20.  $900, 

12$,  22  mo.  13  d. 

3.  $100, 

7$,    8yr. 

21.  $100, 

6$,    1  yr.  6  mo.  15  d. 

4.  $100, 

8$,    7yr. 

22.  $240, 

5  #,    2  yr.  8  mo.  10  d. 

5.  $100, 

8$,    7  yr.  6  mo. 

23.  $360, 

7$,    3  yr.  8  mo.  20  d. 

6.  $100, 

9$,    6  yr.  8  mo. 

24.  $1200, 

8$,    2  yr.  4  mo.  10  d. 

7.  $274, 

10$,  4yr.  10  mo. 

25.  $810, 

9$,    4yr.  2  mo.  10  d. 

8.  $1200, 

7$,    8  yr.  5  mo. 

26.  $720, 

10$,  5  yr.  4  mo.  24  d. 

9.  $796, 

12$,  6  yr.  9  mo. 

27.  $132.48 

11$,  3yr.  1  mo.  6  d. 

10.  $1126.84. 

,  9$,    4  yr.  4  mo. 

28.  $120.48 

12$,  6  yr.  6  mo.  6  d. 

11.  $964.50, 

8$,    12  yr.  6  mo. 

29.  $600, 

5  $,    4  yr.  4  mo.  5  d. 

12.  $360, 

11$,  11  yr.  5  mo. 

30.  $720.84, 

8$,    12yr.3mo.  10  d. 

13.  $10, 

6$,    1  mo.  12  d. 

31.  $900, 

6$,    7yr.  9  mo.  25  d. 

14.  $10, 

6  $,    6  mo.  24  d. 

32.  $840.80, 

10$,  2  yr.  2  mo.  2  d. 

15.  $100, 

8$,    12  mo.  9  d. 

33.  $270.36, 

9$,    4yr.  8  mo.  12  d. 

16.  $600, 

9$,    20  mo.  10  d. 

34.  $143.33, 

11$,  6  yr.  4  mo.  5  d. 

17.  $240, 

5$,    19  rao.  27  d. 

35.  $360.96, 

12$,  5yr.  6  mo.  10  d. 

18.  $396, 

10$,  25  mo.  8  d. 

36,  $770.21, 

7$,    2  yr.  8  mo.  15  d. 

Find  the  interest  and  amount: 

Principal. 

Rate.              Time. 

Principal. 

Rate.              Time. 

1.  $1080.50,  7$,    1  yr.  9  mo. 

13.  $1248, 

9$,    9  mo.  25  d. 

2.  $420.25, 

8$,    2  yr.  9  mo. 

14.  $840, 

6$,    1  yr.  8  mo.  15  d. 

3.  $960, 

9  $,    3  yr.  4  mo. 

15.  $960, 

7$,    1  yr.  9  mo.  24  d. 

4.  $576.48, 

10$,  3  yr.  6  mo. 

16.  $1296, 

8  $,    2  yr.  3  mo.  9  d. 

5.  $645, 

12$,  5  yr.  10  mo. 

17.  $1080, 

9$,    2  yr.  9  mo.  21  d. 

6.  $1200, 

5$,    6  yr.  3  mo. 

18.  $1800, 

10$,  3  yr.  6  mo.  15  d. 

7.  $1200, 

10$,  12  yr.  6  mo. 

19.  $600, 

11$,  4  yr.  7  rao.  18  d. 

8.  $828, 

6$,    8  mo.  16  d. 

20.  $796, 

12$,  5  yr.  10  mo.  6  d. 

9.  $972.36, 

8$,    17  mo.  18  d. 

21.  $976.28, 

7$,    7  yr.  9  mo.  27  d, 

10.  $600.60, 

10$,  23  mo.  14  d. 

22.  $869.44, 

9$,    8  yr.  4  mo.  17  d, 

11.  $1165.17,  12$,  40  mo.  6  d. 

23.  $1126.56 

1,  11$,  10yr.5mo.  1  d, 

12.  $894, 

7$,    14  mo.  17  d. 

24.  $1295.28,  8$,    13yr.  4mo.  29d, 

308  STANDARD  ARITHMETIC. 

To  find  the  time  between  the  dates  here  given,  follow  the 
method  of  Ex.  1,  page  231. 

Find  the  amount: 

Time. 

From  1872,  Oct.  27,  to  1880,  May  12. 

"  1868,  Sept.  19,  to  1870,  June  1. 

"  1872,  Dec.  31,  to  1879,  Oct.  1. 

"  1839,  Jan.  1,  to  1850,  Dec.  20. 

"  1827,  April  1,  to  1847,  July  28. 

"  1868,  Aug.  31,  to  1879,  Nov.  1. 

"  1829,  Feb.  20,  to  1841,  May  10. 

u  1865,  Mar.  15,  to  1877,  Jan.  15. 

u  1849,  June  19,  to  1869,  April  7. 

u  1877,  Nov.  24,  to  1880,  Nov.  30. 

"  1876,  Sept.  27,  to  1879,  Dec.  9. 

"  1866,  Dec.  8,  to  1871,  May  1. 

"  1875,  Dec.  25,  to  1878,  May  28. 

"  1876,  Mar.  21,  to  1879,  June  30. 


Principal. 

Rate. 

1.  $542, 

7* 

2.  $684, 

8* 

3.  $960, 

&* 

4.  $1100, 

10* 

5.  $1186.20, 

n* 

6.  $1260.48, 

12* 

7.  $1040.25, 

8* 

8.  $1097.76, 

«* 

9.  $976.80, 

7* 

10.  $896.84, 

»* 

11.  $1272.24, 

10* 

12.  $1284.96, 

12* 

13.  $1200, 

11*, 

14.  $989, 

12* 

6°/o  Method. 

To  find  the  interest  of  $1  for  any  given  time  at  6  %  affords  an  excellent  mental 
exercise. 

1.  At  6$  per  annum,  what  is  the  interest  of  $1  for  3  vr.  5  mo. 
7d.? 

Solution. — The  interest  of  $1  at  6^  for  41  mo.  is  41  x  5  mills  =  205  mills, 
and  for  7  d.  it  is  1 1/6  mills,  hence  for  3  yr.  5  mo.  and  7  d.  it  is  206  1/6  mills. 

In  like  manner  find  the  interest  of  $1  for — 

2.  1  yr.  2  mo.  8.  3  yr.  1  mo.  3  d.  14.  8  mo.  2  d. 

3.  5  yr.  8  mo.  9.  4  yr.  1  mo.  10  d.  15.  3  yr.  7  d. 

4.  3  yr.  1  mo.  10.  9  yr.  7  mo.  8  d.  16.  11  yr.  11  mo. 

5.  1  yr.  9  mo.  11.  5  yr.  4  mo.  1  d.  17.  11  mo.  5  d. 

6.  10  yr.  4  mo.  12.  1  yr.  11  mo.  20  d.  18.  1  yr.  4  d. 

7.  4  yr.  6  mo.  13.  6  yr.  6  mo.  14  d.  19.  7  mo.  3  d. 

From  the  interest  of  $1  for  any  given  time  the  interest  of  any  other  sum  may 
be  readily  obtained  for  the  same  time  and  at  the  same  rate  per  cent.  For  some  uses 
this  method  is  preferable  to  any  other.     See  Partial  Payments. 


INTEREST.  309 

308.  To  find  the  Rate;  Principal,  Interest,  and  Time  beings 
given. 

ORAL     EXERCISES. 

1.  If  8<fi  are  paid  for  the  use  of  $2  for  1  year,  what  is  the 
rate  per  cent.  ? 

Analysis. — At  1%  per  annum  $2  (2000)  would  earn  20,  but  since  the  given 
interest  (80)  is  4  times  20,  the  rate  must  be  4  times  1%  or  4$. 

2.  Thirty-six  cents  are  paid  for  the  use  of  $3  for  2  years. 
What  is  the  rate  per  cent,  charged  ? 

Analysis. — At  1  %  per  annum  the  interest  of  $3  (3000)  for  1  year  would  be  30, 
and  lor  2  years  it  would  be  60,  but  since  the  interest  paid  (360)  is  6  times  60,  the 
rate  must  be  6  times  1  %  or  6  %. 

3.  Find  the  rate  per  cent,  if  the  annual  interest  on  $840  is 
$42;  if  on  $850  the  interest  is  $34;  if  on  $725  the  interest  is 
$43  y2 ;  if  on  $75  the  interest  is  $3. 

Find  the  rate : 

Principal.       Interest.  Time.  Principal.        Interest.         Time. 

4.  $200,        $18,        lyr.  7.  $120,  $18,      2%yr. 

5.  $150,        $90,         10  yr.  8.  $2000,        $90,         1  yr. 

6.  $180,        $76,        5  yr.  9.  $4000,         $340,       2  yr. 


SLATE     EXERCISES 


Find  the  rate  per  cent. : 

Principal.  Interest.  Time.  Time.* 

1.  $960,  $88.80,  1  yr.  6  mo.  15  d.  4  yr.  7  mo.  15  d. 

2.  $796.20,  $171.98,  2  yr.  8  mo.  12  d.  8  yr.  1  mo.  6  d. 

3.  $897.50,  $251.30,  3  yr.  6  mo.  10  yr.  6  mo. 

4.  $1224.72,  $481.04,  5  yr.  7  mo.  10  d.  16  yr.  10  mo. 

5.  $1152,  $403.20,  3  yr.  10  mo.  20  d.  7  yr.  9  mo.  10  d. 

6.  $867.40,  $320.94,  7  yr.  4  mo.  24  d.  2  yr.  5  mo.  18  d. 

7.  $1231.36,  $923.52,  8  yr.  4  mo.  2  yr.  9  mo.  10  d. 

*  Note. — The  "  time  "  given  in  this  column  being  a  multiple  or  aliquot  of  the 

corresponding  "  time  "  in  the  first  column,  the  rate  for  the  second  column  may  be 

readily  found  from  that  of  the  first.     But  it  must  not  be  forgotten  that,  if  the  time 

is  twice  as  long,  the  rate  must  be  only  1/2  as  great  to  produce  the  same  interest,  etc. 

14 


310  STANDARD  ARITHMETIC. 

SLATE    EXERCI SES. 

Applications. — l.  What  is  the  rate  per  cent,  when  $650  yields 
132%  interest  per  annum?  When  $1250  pays  $43%?  When 
$320  pays  $14.40  ? 

2.  What  is  the  rate  per  cent,  per  annum  when  I  pay  $45  for 
the  use  of  $450  for  two  years  ? 

3.  What  rate,  if  I  pay  194%  on  $900  for  2%  years  ? 

4.  Mr.  Pierce  had  $8000  at  4%#,  and  $1000  at  5f0.  At  what 
rate  must  he  loan  both  sums  to  one  person  to  obtain  the  same 
annual  interest  ? 

5.  A  house,  bought  for  $12,500,  paid  $1000  rent.  If  $200 
were  paid  for  taxes  and  repairs,  what  rate  of  interest  did  the 
purchase  money  yield  ? 

6.  At  what  rate  of  interest  will  $1268  double  itself  in  5  years  ? 

In   16%  years  ?     In  12%  years  ?     (At  what  rate  will  any  sum  double  it- 
self in  the  times  specified  ?) 

7.  Mr.  Williams  borrowed  $8590  on  the  1st  of  June ;  on  the 
25th  of  the  following  January  he  paid  the  amount,  $9036.68. 
What  $  per  annum  did  he  pay  ? 

8.  Mr.  Knell  wishes  to  obtain  a  loan  of  $480,  but  is  not  willing 
to  pay  more  than  |1%  interest  per  month.  What  rate  %  per 
annum  would  that  be  ? 

9.  A  residence  costing  $7500  is  rented  for  $56%  per  month. 
What  rate  per  cent,  per  annum  does  the  money  yield  ? 

10.  Mr.  Dill  had  money  at  3  different  banks  :  in  one  he  had 
$300  at  3$,  in  another  $400  at  4#,  and  in  the  third  $500  at  5#. 
Find  at  what  <f0  he  should  loan  the  three  sums  together  to  get 
the  same  interest. 

11.  Mr.  Ball  put  out  $1200  at  5$  for  6  years.  What  per  cent, 
should  he  charge  to  get  the  same  amount  of  interest  in  5  years  ? 

12.  Seven  months  after  date  a  note  for  $1800  amounted  to 
$1878.75.     What  was  the  rate  ? 


INTEREST.  311 

309.  To  find  the  Time;  Principal,  Interest,  and  Rate  percent, 
being  given. 

ORAL    EXERCISES. 

1.  In  what  time  will  $2  yield  12^  interest,  at  the  rate  of  3$ 
per  annum. 

Analysis. — In  1  year,  at  3  %,  $2  (2000)  yield  60  interest,  and  it  will  require  as 
many  times  1  year  to  produce  120  interest  as  thVe  are  times  60  in  120,  which  is  2. 
Hence  it  will  require  2  years,  etc. 

2.  How  long  will  it  take  $3  to  produce  an  interest  of  54^  at 

6$  per  annum  ? 

Analysis. — In  1  year,  at  6$,  $3  will  produce  an  interest  of  180,  and  it  will  re- 
quire as  many  years  to  produce  540  interest  as  there  are  times  180  in  540,  which 
is  3.     Hence  it  will  require  3  years,  etc. 

3.  In  how  many  years  will  $100  at  4$  pay  $28  interest  ? 
Find  the  time: 

Principal.  Rate.  Interest.  Principal.  Rate.  Interest. 

4.  $200,  4yg&  $45.  7.  $31,     "       50,  $31. 

5.  300  fr.,      4#,  15  fr.  8.  $1200,        5#,  $75. 

6.  £15,  3VS&  £%.  9.  $3400,        70,  $119. 

10.  $50%  interest  is  due  on  $450  at  4yg#.  How  long  has  the 
interest  remained  unpaid  ? 

11.  Mr.  Long  paid  $48  interest.  For  what  period  did  he  pay 
it,  the  principal  being  $640  and  the  rate  5  £  ? 

12.  In  how  many  years  will  the  interest  equal  the  principal  at 
3fo?    At  4%*?    At  6%^? 

(In  what  time  will  $1  principal  produce  $1  interest  at  the  rate  of  3$  a  year?) 


SLATE     EXERCISES 


Find  the  time : 

Principal.  Rate.  Interest.  Principal.  Rate.  Interest. 


1.  $840,  3ys0,  $70.  5.  $4000,  5fc 

2.  $1050,  40,  $136  y2.  6.  $650,  4^ 

3.  320  mark,  4y8&  72  m.  7.  $820,  5$ 

4.  180  lire,  5&  4%  L  8.  $450,  4<£ 


$50.s 

$78.. 

•215  y* 


312  STANDARD  ARITHMETIC. 

Applications. — 1.  Mr.  Hill  invested  $3000  in  government  bonds, 
bearing  5%£  interest.  In  how  many  years  will  the  interest  have 
increased  this  sum  to  $4320  ? 

2.  $600  were  put  at  interest  at  3*/$^,  and  $750  at  4j&  For 
what  time  were  they  loaned,  if  both  sums  together  with  interest 
amount  to  $1525  ? 

(How  much  does  the  first  pay  per  year  ?     The  second  V     Both  ?) 

3.  In  what  time  will  $8000,  invested  at  4y2$,  produce  an  in- 
terest of  $2400  ?  How  long  will  it  be  until  %  of  the  interest  will 
amount  to  as  much  as  the  principal  ? 


Find  the  time 

Find  the  time  i 

Principal. 

Rate. 

Interest. 

Principal. 

Rate. 

Interest. 

4.  $896, 

«fc 

$80.64. 

11.  $998.52, 

5$, 

$185,145. 

5.  $768, 

i£ 

$144,853. 

12.  $1092.24, 

T& 

$338,958. 

6.  $984, 

H, 

$288.64. 

13.  $1129.32, 

»fc 

$582,729. 

7.  $645.75, 

Mi 

$206.64. 

14.  $1192.80, 

8& 

$751,464. 

8.  $727.35, 

12  #, 

$418,954. 

15.  $1200, 

6& 

$1200. 

9.  $866.40, 

ft  & 

$347,065. 

16.  $1268.40, 

12* 

$1268.40. 

10.  $978.60, 

100, 

$518,658. 

17.  $1288.88, 

10* 

$1261.142 

18.  What  would  be  the  effect  on  the  time  if  the  rates  in  prob- 
lems 4-17  were  doubled,  etc.,  etc.?  If  they  were  y3  as  great  as 
those  given  ? 

310.  To  find  the  Principal;  the  Interest,  Rate  per  cent,  and 
Time  being  given. 

ORAL     EXERCISES. 

l.  What  principal  at  5$  will  pay  a  yearly  interest  of  45^  ? 

Solution, — $1  at  5$  will  yield  an  annual  interest  of  50,  and  it  will  require  as 
many  dollars  to  produce  an  interest  of  450  as  there  are  times  50  in  450,  which  is  9. 
Hence  it  will  require  $9,  etc.,  etc. 

/ 


2.  What  principal  at  6$  will  produce  36^  interest  in  2  years  ? 

Solution. — $1  in  2  years  at  <6%  will  produce  an  interest  of  120,  and  it  will  re- 
quire as  many  dollars  to  produce  an  interest  of  360  as  there  are  times  120  in  360, 
which  is  3.     Hence  it  will  require  $3,  etc. 


INTEREST. 


313 


3.  Find  the  principal  which  pays  per  annum  $40  at  5$  ;  $53% 
at  5$;  $100  at  3ys£ 

4.  What  principal  at  5$  will  in  one  year  yield  $30  interest? 

$50?  $60?  $80?   $90? 

5.  What  sum  will  pay  $368  interest  in  20  years  at  5fo  per 
annum  ? 

6.  What  is  the  principal,  if  at  4$  it  pays  $387  interest  in  25 
years  ? 

7.  Find  the  principal  that  will  yield  $441  at  3  y3  <f>  in  30  years, 
and  one  which  will  yield  $111  at  21/2fo  in  40  years. 


SLAT  E 

EX  E  R  C  1  S  E  S . 

Find  the 

principal : 

Find  the 

principal : 

Eate. 

Time. 

Interest. 

Eate. 

Time. 

Interest. 

8. 

H, 

6  months, 

$6.25. 

13.    5fe, 

3  months, 

$61.00. 

9. 

±7iX, 

6      M 

$4.50. 

14.  4%& 

3      " 

$20.25. 

10. 

H, 

6      " 

$27.00. 

15.  6#, 

3      " 

$25.50. 

11. 

H, 

1  month, 

$1.50. 

16.  t%  *, 

1  month, 

$3.00. 

12. 

H, 

1      " 

$1.80. 

17.  60, 

1      " 

$7.50. 

18.  How  many  times  greater  is  the  principal  than  the  annual 
interest,  when  the  rate  is  2  i  ?  2  %  <f0  ?  2  %  0  ?  3  y8  £  ?  3  y3  ^  ? 
4^?    4%^?    5^?    5%^?     6%^?     6%^?     7%^? 

19.  What  is  the  principal  which  yields  $41  interest  per  year  at 
§%*?    At3y80?    At  6%^?    At  2%^? 

20.  What  sum  yields  $47. 25^interest  per  year  at  4%jtf  ? 


F//7tf  £/?e  principal : 


Find  the  principal : 


Rate. 

Time. 

Interest. 

Rate. 

Time. 

Interest. 

2i.  3ys& 

1  year, 

$45  y2. 

27.   5& 

7  years, 

$29.75. 

22. 5y8& 

1    " 

$41  y4. 

23. 3y3& 

4%  " 

$94.50. 

23.  4%^, 

%  " 

$25  /2. 

29.  4#, 

1%  " 

$68.25. 

24.  3%*, 

V." 

$3%. 

3o.  *yfy, 

1%  " 

$47.25. 

25.  8^, 

3/      « 

A 

$18. 

31.  6#, 

573  " 

$170.00. 

26. *%*, 

6    " 

$52%. 

32.  3%& 

4%  " 

$136.00. 

314 


STANDARD  ARITHMETIC. 


SLATE     EXERCISES. 


Find  the  principal 


Interest. 

Rate. 

Time. 

L  $42.70, 

7$, 

From 

Jan.  1,  1880,  to  Sept.  1,  1881. 

2.  $197.80, 

H, 

u 

Jan.  1,  1880,  to  July  12,  1882. 

3.  $26.08, 

H, 

a 

Jan.  1,  1880,  to  Sept.  9,  1882. 

4.  $60.75, 

5£ 

a 

Jan.  1,  1880,  to  Oct.  10,  1881. 

5„  $97,875, 

9$, 

n 

Jan.  1,  1880,  to  July  1,  1881. 

6.  $366.32, 

10$, 

a 

Jan.  1,  1880,  to  Oct.  18,  1883. 

7.  $90.06  + 

11  & 

it 

Jan.  1,  1880,  to  July  1,  1882. 

8.  $561.56, 

12$, 

" 

Jan.  1,  1880,  to  Oct.  1,  1884. 

9.  $445.19, 

*% 

u 

Jan.  1,  1880,  to  July  24,  1885. 

10.  $277.76, 

8$, 

(i 

Jan.  1,  1880,  to  Nov.  15,  1883. 

11.  $315.64  + 

5$, 

a 

Jan.  1,  1880,  to  Aug.  6,  1885. 

12.  $95.97, 

6$, 

a 

J;in.  1,  1880,  to  Nov.  1,  1882. 

13.  $700.70, 

9$, 

a 

Jan.  1,  1880,  to  Oct.  10,  1889. 

14.  $1150.86, 

12$, 

(( 

Jan.  1,  1880,  to  July  20,  1887. 

Applications. — l.  Mr.  Day  wishes  to  invest  snch  a  sum  that  at 
5  $  he  may  draw  $10,000  interest  in  20  years.  What  sum  must 
he  invest  ? 

2.  The  interest  for  1  %  years  of  a  principal  drawing  3  y3  $  in- 
terest per  annum  is  paid  with  7  bu.  of  wheat  at  $llfu  a  bu.,  and 
30  bu.  of  potatoes  at  $1  a  bu.     What  is  the  principal  ? 

3.  Mr.  A.  has  invested  $5500  at  5  $  interest.  Mr.  B.  receives 
for  his  money  $25  more  interest,  though  invested  at  1%$L  less 
than  A.'s.     What  is  B.'s  principal  ? 

4.  Mr.  Jones  proposes  to  buy  the  house  in  which  he  resides. 
The  rent  he  pays  is  $540  per  year.  He  does  not  wish  to  pay 
more  for  interest,  insurance,  taxes,  and  repairs,  than  he  now  pays 
for  rent.  What  sum  can  he  offer  for  the  house,  if  he  allows  4%J< 
for  interest  and  l3/4$  for  taxes  and  repairs  ? 

5.  What  may  I  offer  for  a  house  which  pays  $1350  rent  per 
year,  so  that  the  money  I  invest  may  bear  7%^  interest  ? 


INTEREST.  315 

311.  To  find  the  Principal;  the  Amount,  Time,  and  Rate  per 
cent,  being  given. 

1.  What  principal  at  5$  will  produce  an  amount  of  $2.10  in 

1  year  ? 

Solution. — $1  at  5  %  will  amount  to  $1.05  in  1  year,  and  to  produce  an  amount 
of  $2.10  in  the  same  time  will  require  as  many  dollars,  principal,  as  there  are  times 
$1.05  in  $2.10,  or  $2. 

2.  What  principal  at  6$  will  amount  to  $3.54  in  3  years  ? 

Solution. — $1  at  &%  will  amount  to  $1.18  in  3  year3,  and  to  produce  an  amount 
of  $3.54  in  the  same  time  will  require  as  many  dollars,  principal,  as  there  are  times 
$1.18  in  $3.54,  or  $3. 

3.  What  sum  of  money  must  be  put  to  interest  at  6$  to 
amount  to  $5.30  in  1  year  ?  To  $6.72  in  2  years  ?  To  $3.72  in 
3  years  ? 

4.  Find  the  principal  which  will  in  two  years  amount  to  $8. 72 
at  4%$  per  annum. 


ORAL 

AND      SLATE      EXERCISES. 

What  sum  must  be  put  at  in\ 

teres t  for 

5.  2  yr.  at  4$  to  amt.  $5.40  ? 

11.  2%; 

yr.  at  2$  to  am 

t.  $5.30? 

6.  4      "     6f0     " 

$12.40? 

i2. 3y3 

"     6f0     " 

$12.00  ? 

7.  6      "     2%^" 

$9.20? 

13.   7% 

"     8f0     " 

$8.00  ? 

8.  3      "     3fo     " 

$8.72? 

14.  4% 

"       Sfo       " 

$4.51  ? 

9.  10     "      7fo      " 

$3.40? 

15.   9% 

"       lfo       " 

$8.78? 

10.  8      "     hfo     " 

$15.40? 

16.  6% 

"     5fo     " 

$12.00? 

17.  What  sum  must  Mr.  Day  invest,  that  it  may  amount  to 
$10,000  in  20  years  at  5?0  ? 

18.  An  investment  at  5  fo  made  by  Mr.  Palmer  15  years  ago, 
now  amounts  to  $1984.40.     What  was  the  sum  invested  ? 

19.  Ten  years  after  buying  a  city  lot,  Mr.  D.  sold  it  for 
$18,678,  thereby  obtaining  interest  at  the  rate  of  12$  per  annum 
on  the  sum  paid  for  it.     What  was  the  purchase  price  ? 

20.  What  principal  at  interest  for  4  years  at  8$  will  amount 
to  the  same  as  $3500  at  4$  in  the  same  time  ? 


316  STANDARD  ARITHMETIC. 

Present  Worth. 

312.  The  present  worth  of  a  debt,  payable  at  a  future  time 
without  interest,  is  such  a  sum  of  money,  that,  if  put  at  interest, 
it  would  amount  to  the  debt  when  it  becomes  due. 

The  problem  in  present  worth  is  similar  to  the  preceding  one,  that  is,  it  is  re- 
quired to  find  the  principal,  the  amount,  time,  and  rate  per  cent,  being  given. 

313.  The  difference  between  the  face  of  a  debt  and  its  present 
worth  is  True  Discount. 

True  discount  is  so  called  to  distinguish  it  from  bank  discount.  True  discount 
is  the  interest  on  the  present  worth  of  a  debt.  Bank  discount  is  interest  on  the 
debt.     The  difference  is  the  interest  on  the  true  discount. 

Examples. — l.  Find  the  present  worth  and  the  true  discount 

of  a  debt  of  $48.36  payable  in  6  months  without  interest. 

Analysis. — Since  one  dollar,  at  interest  for  six  months,  will  at  the  end  of  the 
time  amount  to  $1.03,  every  dollar  and  three  cents  in  $48.36,  due  at  the  end  of  six 
months,  is  worth  one  dollar  now ;  hence  the  solution  $48.36  -i-  $1.03  =  46.951. 

2.  Find  the  present  worth  of  $59.50  payable  in  3  mo.  without 
interest. 

3.  Find  the  present  worth  of  $118.20  payable  in  8  mo.  without 
interest. 

4.  What  is  a  debt  of  $100  worth  now,  which  in  1  month  is  to 
be  paid  without  interest  ?    Suppose  the  debt  to  be  $48.35. 

5.  Find  the  true  discount  of  a  debt  of  $763.30  due  in  9  months 
and  bearing  no  interest. 

6.  Find  the  true  discount  of  a  debt  of  $364.48  due  without  in- 
terest in  6y2  months  ;  also  of  $37.44  due  in  8  months  without  in- 
terest. 

7.  Mr.  Hall  owes  me  $968,  due  in  two  months  without  interest. 
If  he  desires  to  pay  me  now,  what  sum  should  I  accept  if  the  use 
of  the  money  is  worth  8  #  per  annum  ? 

8.  What  is  the  present  worth  of  the  amount  which  will  be  due 
on  a  note  for  $3500  having  1  yr.  2  mo.  18  d.  to  run,  at  6  $,  if 
the  current  rate  of  interest  is  12  #  ? 


INTEREST. 


317 


Exact  Interest. 

3I4-.  In  all  the  foregoing  calculations  30  days  have  been  taken  for  one  month, 
and  360  days  for  one  year.  According  to  this  method  the  interest  on  $6000  for 
30  days  at  6  %  would  be  $30 ;  but  if  the  period  of  30  days  were  reckoned  as  only 
30/365  of  a  year,  the  interest  would  be  $29.52.  The  latter  is  called  Accurate  or 
Exact  Interest. 

315.  Accurate  or  Exact  Interest  is  interest  for  the  exact 
time  in  days,  the  days  being  reckoned  as  365ths  of  a  year. 

The  United  States  Government  pays  exact  interest. 

316.  To  obtain  accurate  or  exact  interest  for  any  number  of 
days,  we  find  the  exact  time  in  days,  and  take  as  many  365ths 
of  a  year's  interest  as  there  are  days  in  the  given  time. 

Example. — l.  $350  is  at  interest  from  January  16  to  March  29, 
1880.     What  is  the  accurate  interest  ? 


Exact  Timei 

Solution. 

From  Jan.  16  to  31 

=  15d.            $3.50  =  int.  at  1$ 

for  1  yr. 

In  February 

29  d.                    6 

To  March  29 

29  d.            21.00  =  int.  at  6f0 

for  1  yr. 

73~d.            73  x  $21  _  $1  30  . 

=  int.  at  6 

365 

Principal. 

Time. 

Rate. 

2.  $5000, 

From  Jan.  1  to  Aug.  16,  1875, 

5£ 

3.  $16500, 

u 

Feb.  10  to  Sept.  15,  1879, 

•£ 

4.  $24800, 

u 

Mar.  8  to  Oct.  3,  1879, 

6%. 

5.  $7500, 

u 

July  2  to  Nov.  9,  1880, 

5<fc. 

6.  $9800, 

u 

May  5  to  Dec.  1,  1879, 

7fo. 

7,  $187.44, 

u 

Jan.  20  to  April  U,  1881, 

V3/10* 

8.  $768, 

u 

Jan.  24  to  July  30,  1877, 

6£ 

9.  $840, 

a 

June  15  to  Oct.  25,  1878, 

7f0. 

10.  $480, 

U 

Mar.  31  to  Oct.  4,  1866, 

8/c 

11.  $1080, 

u 

Jan.  1  to  July  16,  1867, 

5£ 

12.  $975, 

a 

Sept.  15  to  Dec.  30,  1852, 

9£ 

13.  $1104, 

u 

Feb.  28  to  Aug.  31,  1870, 

10£ 

14.  $1020, 

a 

April  5  to  Oct.  26,  1874, 

7fo. 

15.  $1121, 

M 

Sept.  24  to  Dec.  21,  1872, 

6£ 

for  73  d„ 


318  STANDARD  ARITHMETIC. 

Common  Business  Method. 

317.  Bankers  and  most  other  business  men,  in  computing 
interest  for  short  periods  of  time,  usually  take  the  exact  number 
of  days  as  above,  but  reckon  each  day  as  y360  of  a  year. 

Days  of  Grace. — Formerly  granted  as  a  favor,  the  custom  of  allowing  three 
days  beyond  the  time  agreed  upon  for  the  payment  of  a  debt  has  grown  into  a  very 
general  law.  They  are,  however,  of  no  advantage  to  the  debtor,  inasmuch  as  the 
interest  continues  up  to  the  time  of  payment.  In  States  where  they  are  required, 
suit  can  not  be  brought  to  enforce  the  payment  of  a  debt  till  the  expiration  of  the 
days  of  grace. 

Note. — In  the  following  problems,  add  days  of  grace  to  the  given  time. 

Example.— l.  Find  the  interest  of  $150  from  April  27  to  June 

26,  1885,  at  90. 

In  April,     3  d.  1.50          Int.  for  60  d.  at  60 

In  May,     31  d.  75                        3     "      " 

In  June,    26  d.  2)1.575                      ~63     "       " 

Grace,         3  d.  ~~  .787  V* 

63  d.  2.362  Vg  Int.  for  63  d.  at  90 

Find  the  interest  of 

Principal.  Time.  Rate. 

2.  $450,  From  Aug.  10  to  Nov.  8,  1885,  60. 

3.  $720,  "  Jan.  25  to  April  7,  1885,  70. 

4.  $960,  "  Feb.  3  to  Mar.  19,  1884,  80. 

5.  $540,  "  April  8  to  May  18,  1870,  90. 

6.  $100,  "  Jan.  30  to  Mar.  6,  1872,  40. 

7.  $900,  "  Feb.  12  to  Mar.  4,  1873,  7  */,  0. 

8.  $240,  "  May  31  to  Nov.  27,  1875,  100. 

9.  $333,  "  Aug.  1  to  Nov.  29,  1876,  5  0. 

10.  $672,  "  Feb.  28  to  Oct.  25,  1880,  4  */i  0. 

11.  $60,  "  June  19  to  Nov.  10,  1881,  120. 

12.  $600,  "  July  4  to  Oct.  20,  1882,  30. 

13.  $630,  "  Feb.  1  to  Aug.  20,  1883,  5  */7  0» 

14.  $480,  "  Jan.  21  to  Dec.  2,  1871,  50. 

15.  $270,  "  May  10  to  July  29,  1874,  60. 

16.  $386,  "  Oct.  13  to  Dec.  12,  1885,  90. 


INTEREST.  319 

Bank  Discount. 

Illustration. — If  a  merchant  desires  to  obtain  a  loan  for  any  short  period  of 
time,  say  about  $1000,  for  60  days,  he  may  take  his  own  note,  or  the  note  of  an- 
other party,  for  $1000,  to  a  bank,  and  if  the  note  is  properly  indorsed  or  its  pay- 
ment is  otherwise  secured,  the  bank  will  take  it,  pay  him  $1000  less  the  interest  for 
63  days,  and  collect  the  $1000  when  it  becomes  due. 

The  interest  on  $1000  for  60  days  +  3  days  of  grace,  computed  by  any  of  the 
preceding  methods,  is  $10.50.  $1000  —  $10.50  =  $9S9.50  =  the  sum  which  the 
merchant  will  receive  on  the  note. 

318.  The  Avails  or  Proceeds  of  a  note  is  the  sum  that  the 
bank  pays  upon  it  after  deducting  the  interest  on  the  face  or 
amount  of  the  note. 

319.  The  sum  deducted  for  the  payment  of  proceeds  before  a 
note  becomes  due  is  called  Bank  Discount. 

320.  A  note  is  said  to  be  payable  at  the  promised  time  of 
payment.  Its  date  of  maturity  is  three  days  later,  the  last  day 
of  grace.     It  then  becomes  legally  due. 

If  a  note  is  not  paid  at  maturity,  a  written  notice  of  the  failure  should  be  sent 
on  the  last  day  of  grace  to  the  indorscr  or  indorsers,  otherwise  they  can  not  be  held 
for  its  payment.  Such  notice  is  called  a  protest,  and  should  be  made  out  by  a 
Notary  Public. 

321.  The  dates  showing  when  a  note  is  payable  and  the  date 
of  maturity  are  written  thus  :  Aug.  5/8. 

322.  If  the  third  day  of  grace  falls  on  a  Sunday  or  on  a  legal 
holiday,  the  note  is  due  the  day  before. 


SLATE     EXERCISES. 

Example. — l.  What  was  the  bank  discount  and  what  were  the 
proceeds  of  a  note  of  $645,  dated  July  22,  payable  Sept.  20,  and 
discounted  at  6  #  on  the  day  the  note  was  drawn  ? 

Due  Sept.  20/23-  Discount. 

In  July  =    9  d.  $6.45     Int.  60  d.  $645        Face. 

In  Aug.  =  31  d.  .32       "      3  d.  6.77 

In  Sept.  =  23  d.  (3  d  grace)       $6.77      "    63  d.  $638.23  Proceeds. 

63  d. 


320  STANDARD  ARITHMETIC. 

Find  the  bank  discount  and  proceeds  of  a  note  for 

2.  $440,  payable  in  90  d.,  discounted  at  6f0  on  the  day  drawn. 

3.  $500,         "         .60  d.,  "  9$     "  " 

4.  $1000,       "  45  d.,  "  5fc      " 

5.  $140.25,    "  30  d.,  "  4*/**" 

6.  $970,  dated  Feb.  9,  1884,  payable  in  60  days,  and  discounted 

at  6$  on  March  13,  1884.      (Due  April  9/12.     Keckon  discount  from  March 
13th  to  April  12th  =  30  d.) 

7.  $638,  dated  Jan.  21,  1880,  payable  in  3  mo.,  discounted 

at  8$  on  Feb.  4,  1880.      (Due  April  21/24.     Discounted  for  80  d.) 

8.  $1235,  dated  May  10,  1881,  payable  in  4  mo.  with  interest 
at  6$,  discounted  at  6$  on  June  5,  1881.     (Due  Sept.  10/13,  with  in- 

terest  for  4  mo.  3  d.     The  amount  is  the  sum  to  be  discounted.) 

9.  $6040,  dated  July  12,  1885,  payable  in  90  d.  with  interest 
at  4}/2?c,  discounted  at   10$   on  Aug.  4,  1885.      (Due  Oct.  10/13. 

Reckon  interest  for  93  d.  at  4  1/2  %  and  discount  from  Aug.  4th  to  Oct.  13th.) 

10.  $12008,  dated  May  12,  1870,  payable  in  9  mo.  with  in- 
terest at  Sfc ;  indorsed  $5000,  Aug.  12,  1870,  discounted  Nov.  12, 
1870.      (Due  Feb.  12/15.) 

Find  interest  on  $12008  for  3  mo.  at  8$;  apply  the  partial  payment,  $5000, 
and  find  new  principal ;  reckon  amount  of  this  new  principal  for  6  mo.  3  d. ;  on 
this  amount  reckon  discount  from  Nov.  12,  1870,  to  Feb.  15,  18*71  ==  95  d. 

11.  Find  the  face  of  a  note,  payable  in  57  d.,  that  will  yield 

$792  proceeds  when  discounted  at  6$. 

Analysis. — Since  a  face  of  $1  would  yield  $.99  proceeds,  to  yield  $792  pro- 
ceeds will  require  as  many  dollars  face  as  $.99  are  contained  times  in  $792  = 
800  times  $1  =  $800. 

12.  Find  the  face  of  a  note,  payable  in  30  days,  that  will  yield 
$477.36  proceeds  when  discounted  at  6$. 

13.  I  bought  a  bill  of  goods  for  $864  on  4  mo.  credit,  but, 
being  offered  5  $  off  for  cash,  I  borrowed  the  money  at  a  bank  by 
having  my  note,  payable  in  117  days,  discounted  at  6$,  and  paid 
the  bill.  What  was  the  face  of  the  note,  and  how  much  did  I 
gain  ? 


INTEREST.  321 

Promissory  Notes. 

323.  A  Promissory  Note,  or  simply  a  Note,  is  a  written 
promise  to  pay  a  specified  sum  of  money  on  demand  or  at  a  cer- 
tain time  to  some  person  named  in  the  note. 

324.  The  sum  promised  to  be  paid  is  the  Face  of  the  note. 
326.  The  signer  is  called  the  Maker  or  Drawer. 

326.  The  person  to  whom  it  is  payable  is  the  Payee. 

327.  The  owner  of  the  note  is  called  the  Holder. 
The  following  is  a  simple  form  of  a  promissory  note  : 

$50.75.  New  York,  June  15,  1880. 

I  Three  months  \  after  date  I  promise  to  pay  \  John  Jones  2 1 

\fifty  75/ioo  dollars,*  \  for  value  received. 


Amos  Wilson. 


(Who  is  the  maker  of  the  above  note  ?  Who  the  payee  ?  What  is  its  face  ? 
What  is  its  date  ?) 

Notes. — 1.  Time.  a.  The  time  of  a  note  should  be  written  in  words,  and  may 
be  expressed  in  days,  months,  or  years,  or  the  date  on  which  it  is  to  be  paid  may 
be  specifically  stated.  Thus,  "  on  the  first  day  of  May,  1887, 1  promise"  etc. 
Notes  in  which  a  certain  time  is  given  for  payment  are  called  Time  Notes. 

b.  If,  in  place  of  "sixty  days  after  date"  the  words  "on  demand"  were  used, 
the  note  would  be  payable  on  demand.     It  would  then  be  called  a  "  demand  note." 
2.  Payee,  a.  As  it  now  stands,  the  note  is  payable  only  to  Mr.  John  Jones  in 
person.     A  note  thus  drawn  can  not  be  transferred,  and  is  said  to  be  "  not  negoti- 
able." 

b.  But  if  the  note  read,  to  "  John  Jones  or  order"  Mr.  Jones  could  make  it 
payable  to  whomsoever  he  might  order.     Thus,  if  he  wrote 

Pay  to  William  Jackson, 
John  Jones, 
then  William  Jackson  would  be  the  only  person  to  whom  it  would  be  payable.    But 
if  Mr.  Jones  wrote 

Pay  to  William  Jackson,  qr  order, 
John  Jones, 
then  Mr.  Jackson  could  in  his  turn  make  it  payable  to  whomsoever  he  pleased,  and 
in  this  way  it  might  pass  through  dozens  of  hands. 


STANDARD  ARITHMETIC. 

c.  If  the  words  "  or  bearer "  followed  Mr.  Jones's  name,  it  would  be  payable 
to  any  one  who  might  present  it.  Or,  if  Mr.  Jones  should  write  only  his  name  on 
the  back  it  would  then  be  payable  to  any  one  who  might  hold  it,  as  if  made  payable 
to  bearer.     Such  an  indorsement  is  called  an  indorsement  in  blank. 

d.  A  note  that  may  be  transferred  from  one  person  to  another,  either  by  deliv- 
ery or  indorsement,  is  said  to  be  negotiable. 

3.  The  face.  The  number  of  dollars  for  which  the  note  is  drawn  should  be 
specified  in  words,  the  cents  may  be  expressed  as  hundredths  of  a  dollar. 

4.  Interest.  The  note  as  printed  would  not  bear  interest  till  due.  If  not  then 
paid  it  would  thereafter  be  subject  to  legal  interest.  If  the  words  "  with  interest " 
were  used,  and  no  rate  specified,  the  note  would  draw  interest  at  the  legal  rate  from 
date  to  time  of  payment.  If  the  rate  were  any  other  than  the  legal  rate  it  would 
have  to  be  specified,  as,  "with  interest  at  8#." 

5.  The  place  of  payment.  If  the  words,  at  the  First  National  Bank,  or  other 
place,  were  written  after  the  "fifty  75/i0o  dollars,"  then  the  note  should  be  pre- 
sented at  the  bank  or  other  place  named,  on  the  last  day  of  grace.  If  no  bank  or 
other  place  of  payment  is  specified,  the  note  is  payable  at  the  drawer's  place  of 
business  or  residence. 

6.  If  the  words  "  value  received  "  are  omitted,  the  holder  may  have  to  prove 
that  the  drawer  received  its  value. 


EXERCISES    IN     WRITING     NOTES. 

1.  Draw  a  note  payable  by  John  Doe  to  Richard  Eoe  ;  another 
to  Richard  Roe  or  order  ;  another  to  Richard  Roe  or  bearer. 

2.  Draw  a  demand  note  ;  a  time  note.     Draw  a  time  note  with 

interest  at  legal  rate.      (The  words  with  interest  mean  as  much  as  with  interest 
at  legal  rate.) 

3.  Draw  a  note  in  which  the  date  of  payment  is  specifically 
stated. 

4.  Draw  a  time  note  in  which  the  rate  of  interest  is  other  than 
the  legal  rate. 

5.  Draw  a  time  note  and  indorse  it  in  blank. 

6.  Draw  a  demand  note,  payable  to  Henry  Hudson,  or  order, 
and  properly  indorse  it  to  Miles  Standish,  or  bearer. 

7.  Draw  a  note  to  mature  3  months  from  the  present  date. 

8.  Draw  a  note  payable  to  order  for  any  given  sum,  specifying 
the  place  and  time  of  payment. 


INTEREST.  323 

Partial  Payments. 

328.  Partial  payments  are  payments  in  part  of  any  indebt- 
edness. It  is  customary  to  indorse  partial  payments  on  the  back 
of  a  note.  The  indorsement  consists  of  the  date  and  amount 
of  the  payment. 

Example. — l. 

$100.  Cincinnati,  0.,  May  10,  1876. 

For  value  received,  on  demand,  I  promise  to  pay  to  Warren 
Hastings,  or  order,  one  hundred  dollars  with  interest  at  6$. 

Robert  Moulton. 

On  this  note  partial  payments  were  indorsed  as  follows  :  Nov. 
10,  1876,  $23  :  May  10,  1879,  $17  ;  May  10,  1881,  $7  ;  Sept.  10, 
1882,  $33.     What  was  the  amount  due  Jan.  10,  1883  ? 

Solution. . 

Principal  from  May  10,  1876 $100 

Interest  to  Nov.  10,  1876  (6  months) +3 

Amount 103 

First  payment,  Nov.  10,  1876 —23 

New  principal 80 

Interest  on  $80  to  May  10,  1879  (2  1/2  years) +12 

Amount 92 

Second  payment,  May  10,  1879  ...  —17 

New  principal 75 

Interest  on  $75  to  May  10,  1881  (2  years) ($9) 

(Here  the  payment  ($7)  is  less  than  the  interest,  and  if  we  were  to 
form  a  new  principal,  as  in  the  cases  preceding  this,  it  would 
be  equivalent  to  adding  the  unpaid  interest  ($2)  to  the  principal. 
But,  this  being  illegal,  we  continue  the  interest  on  $75  till  the 
sum  of  the  payments  equals  or  exceeds  the  interest ;  hence  we 
find  the) 

Interest  on  $75  from  May  10,  1879,  to  Sept.  10,  1882 15 

Amount $90 

Third  and  fourth  payments  ($7  +  $33) —40 

principal 60 

Interest  on  $50  to  Jan.  10,  1883  (4  months) _-f-l 

Amount  due  on  settlement,  Jan.  10 $51 


324 


STANDARD  ARITHMETIC. 


This  calculation  is  in  accordance  with  the 

329.    United   States  Rule  for  Partial    Payments. 

Find  the  amount  of  the  principal  to  the  time  when  a  payment 
or  the  sum  of  two  or  more  payments  equals  or  exceeds  the  inter- 
est. From  this  amount  deduct  the  payment  or  the  sum  of  the 
payments. 

With  the  remainder  as  a  new  principal,  proceed  as  before. 

Note. — This  rule  is  founded  on  the  principle  that  neither  interest  nor  payment 
shall  draw  interest. 

Examples.— 2.  A  note  of  $250  is  dated  June  1,  1878.  Indorse- 
ment :  June  1,  1879,  $85.  What  was  due  at  the  time  of  settle- 
ment, June  1,  1880,  interest  at  5$  ? 

3.  A  note  of  $1000,  dated  May  1,  1875,  was  indorsed  as  follows : 
Nov.  25,  1875,  $134;  March  7,  1876,  $315.30;  Aug.  13,  1876, 
$15.60 ;  June  1,  1877,  $25  ;  April  25,  1878,  $236.20.  What  was 
the  amount  due  on  Sept.  10,  1878,  interest  6  <f0  ? 

When  there  are  many  dates  to  deal  with,  as  in  the  preceding  problem,  it  will 
aid  the  accountant  to  avoid  confusion  and  consequent  danger  of  error  to  write  out 
the  dates  as  in  the  first  column  below,  and  the  differences  as  in  the  second.  This 
arrangement  affords  a  ready  means  of  testing  the  accuracy  of  the  work,  inasmuch 
as  the  sum  of  the  differences  in  the  second  column  must  be  equal  to  the  difference 
between  the  first  and  last  dates  in  the  first. 

Then,  if  we  adopt  the  6  %  method  (see  page  308),  that  is,  multiply  the  interest 
of  $1  for  each  period  by  the  number  of  dollars  at  interest  during  that  period,  we 
may  test  the  correctness  of  each  item,  since  the  sum  of  the  several  items  must  equal 
the  interest  of  $1  for  the  whole  time. 


Dates. 

Time 
elapsed. 

Int.  of  ft. 

Principal. 

Int.  of 
principal. 

Amount. 

Payment. 

Yr.          mo.     d. 

Mo.     d. 

1875,     5,     1 

1875,  11,  25 

6,  24 

$.034 

$1000 

$34 

1034 

134 

1876,     3,     7 

3,   12 

.017 

900 

15.30 

915.30 

315.30 

1876,     8,  13 

5,     6 

.026 

600 

15.60 

615.60 

15.60 

1877,     6,     1 

9,  18 

.048 

600 

28.80  ) 
32.40  J 

661.20 

25.      ) 

1878,     4,  25 

10,  24 

.054 

600 

236.20  j 

1878,     9,  10 

4,  15 

.0225 

400 

9 

409 



3,     4,     9 

40,     9 

.2015 

$409  balance  remaining  unpaid. 

INTEREST.  325 

4.  A  note  of  $600,  dated  March  10, 1877,  is  indorsed  as  follows : 
Sept.  10,  1877,  $100 ;  June  10,  1878,  $100 ;  Dec.  10,  1878,  $100 ; 
Dec.  10,  1879,  $200.  What  amount  was  due  on  Oct.  10,  1880,  at 
the  time  of  settlement,  interest  6fo  ? 

5.  I  held  a  note  for  $600,  bearing  interest  at  6  <f>  from  March 
8,  1869.  I  received  as  partial  payments  (1)  $140  on  Sept.  10, 
1870 ;  (2)  $50  on  July  20,  1872.  What  amount  was  due  on  set- 
tlement, Oct.  15,  1873  ? 

6.  A  note  of  $300,  dated  Jan.  1,  1878,  had  the  following  in- 
dorsements :  Aug.  1,  1878,  $50 ;  Dec.  1,  1878,  $50 ;  July  1,  1879, 
$100.     What  amount  was  due  on  Jan.  1,  1880,  interest  at  7$  ? 

7.  On  the  1st  of  June,  1879,  H.  R.  Fox  borrowed  of  Charles 
Lever  $800,  and  gave  his  note  for  that  sum  with  interest  at  7$. 
Sept.  1,  1879,  Fox  made  a  payment  of  $240,  and  a  new  note  was 
made  out  for  the  balance.     What  was  the  face  of  this  note  ? 

(Write  this  new  note  out  in  proper  form,  dating  it  at  Richmond,  Va.) 

8.  -$575.  Cleveland,  O.,  Feb.  1,  1879. 
Eight  months  after  date  I  promise  to  pay  C.  F.  Cutter  & 

Co.,  or  order,  five  hundred  seventy-five  dollars,  ivith  interest  at 

7fc,  for  value  received. 

R.   W.   Cane. 

Indorsements:  Oct.  1,  1879,  $300;  July  1,  1880,  $150. 

What  balance  was  due  on  settlement,  Dec.  1,  1880  ? 

9.  Aug.  1,  1873,  F.  Critland  gave  to  Robert  Ingham  a  note 
for  $143.50,  with  interest  at  7$.  He  made  payments  as  follows  : 
Dec.  17,  1873,  $37.40;  July  1,  1874,  $7.09;  Dec.  22,  1875,  $13.13  ; 
Sept.  9,  1876,  $50.50.     What  amount  was  due  Dec.  28,  1876  ? 

10.  A  note  for  $2800,  dated  June  30,  1884,  has  the  following 
indorsements  :  Dec.  23,  1884,  $50  ;  May  16,  1885,  $90 ;  June  3, 
1885,  $10  ;  April  23,  1886,  $150  ;  May  30,  1886,  $22.  What  bal- 
ance remained  due  at  the  last  payment  ?    (Why  the  original  principal  ?) 


326  STANDARD  ARITHMETIC. 

The   Mercantile   Rule. 

330.  The  following  method  of  finding  the  balance  due  on  a 

mercantile  account  or  other  debt  running  for  a  year  or  less  is  very 

commonly  adopted  when  partial  payments  have  been  made. 

The  principle  on  which  it  is  based  will  be  readily  understood  from  the  state- 
ment of  the  rule. 

331.  Rule,— l.  Compute  the  amount  of  the  debt  from  its  date 
to  the  time  of  settlement. 

2.  Compute  the  amount  of  each  payment  from  its  date  to  the 
date  of  settlement. 

3.  Subtract  the  sum  of  the  amounts  of  the  several  payments 
from  the  amount  of  the  debt. 

The  difference  will  be  the  balance  due. 

The  answers  given  to  the  following  problems  are  based  on  the  common  method 
of  finding  differences  of  time  (compound  subtraction — 3 GO  days  to  the  year)  though 
other  methods  may.be  used. 


SLATE   EXERCI SES. 

1.  On  a  debt  of  $420,  contracted  March  15,  1885,  payments 
were  made  as  follows  :  June  1,  1885,  $150 ;  Sept.  6,  1885,  $130 ; 
Oct.  14,  1885,  $75.  What  was  the  balance  due  Dec.  24,  1885,  at 
7$  interest  ? 

2.  On  a  note  for  $645,  at  6  $  interest,  dated  Jan.  1,  1886,  and 
maturing  9  months  after  date,  the  following  indorsements  were 
made  :  Mar.  4,  1886,  $50 ;  Apr.  2,  1886,  $75  ;  Aug.  10,  1886, 
$200.     What  was  the  balance  due  at  time  of  payment  ? 

3.  ^Tr.  Thomas  gave  his  note,  dated  Feb.  15,  1885,  to  Amos 
King,  for  $1940,  payable  Jan.  1,  1886,  with  interest  at  Sfc.  Allien 
due,  the  note  had  the  following  indorsements  :  Aug.  1,  1885, 
$440  ;  Sept.  6,  1885,  $500 ;  Oct.  1,  1885,  $300 ;  Nov.  15,  1885, 
$400.     What  was  the  balance  due  Jan.  1,  1886  ? 

4.  May  1, 1884,  goods  were  purchased  to  the  value  of  $1250,  on 
which  the  following  payments  were  made  :  Aug.  1,  1884,  $250 ; 
Sept.  4,  1884,  $300  ;  Oct.  15,  1884,  $450  ;  Dec.  8,  1884,  $120. 
What  was  the  balance  due  Dec.  20,  1884  ? 


INTEREST.  327 

Annual  Interest. 
332.  Annual  Interest  is  simple  interest  payable  annually. 

Iu  some  States  annual  interest,  if  not  paid  when  due,  draws  interest  at  the  same 
rate  as  the  principal,  in  others  at  the  legal  rate,  whatever  may  be  the  rate  paid  on 
the  principal ;  but  in  some  it  is  illegal  to  collect  interest  on  unpaid  annual  interest. 

Example. — l.  Mr.  Hart  gives  his  note  for  $2000,  payable  in  4 
y^ars  with  interest  annually  at  5  $.  What  will  be  the  amount  due 
when  the  note  matures,  provided  Mr.  Hart  pays  interest  as  agreed 
upon  ? 

Solution. — The  interest  on  $2000  at  5%  is  $100  per  year.  Hence  Mr.  Hart 
should  pay  $100  each  year  till  the  close  of  the  4th,  when  he  should  pay  the  last 
year's  interest  ($100)  with  the  principal,  making  together  $2100;  but 

2.  If  Mr.  Hart  does  not  pay  the  interest  annually,  as  agreed, 
his  account  at  the  close  of  the  4  years  would  stand  as  follows  : 

The  principal,  $2000 

Total  annual  interest  ($100  per  year  for  4  yr.)  =  $400 
Interest  on  1st  ann.  int.  (3x5x$l)  =       15 

"        "   2d         "       (2  x  5  x  $1)  =       10 

"   3d  "        (1  x  5  x  $1)  =   5  430 

Amount  due  at  the  close  of  the  4th  year,  =  $2430 

This  amount  differs  from  the  amount  of  the  same  principal  at  simple  interest  only 
by  the  $30  interest  on  the  deferred  payments. 


Applications. — l.  The  annual  interest  on  $2000  invested  at  6  $ 
for  6  years  remaining  unpaid,  what  is  the  amount  due  ? 

2.  I  invested  $550  for  3  years  at  5  $  interest,  payable  annually. 
What  wTas  due  at  the  end  of  the  three  years,  interest  for  the  first 
only  having  been  paid  ? 

3.  At  the  end  of  3  years,  what  is  due  on  a  debt  of  $500,  with 
interest  payable  annually  ? 

4.  Find  the  amount  due  in  8  years  on  $320,  invested  at  7^ 
interest,  payable  annually,  the  interest  for  the  first  3  years  having 
been  paid  when  due. 

5.  The  interest  on  a  note  of  $400,  payable  after  6  years,  with 
annual  interest  at  5*/tA  has  not  been  paid.  What  is  the  note 
worth  at  the  time  of  its  becoming  due  ? 


328  STANDARD  ARITHMETIC. 

6.  A  note  for  $1000,  dated  July  1,  1865,  at  7$  interest,  pay- 
able annually,  was  paid  January  1,  1868.  What  was  the  amount 
due  at  maturity  ? 

7.  No  interest  having  been  paid  on  a  note  for  $500,  dated 
June  1>  1878,  with  interest  payable  annually,  find  the  amount 
due  September  1,  1880. 


Miscellaneous  Problems, 

1.  A  gentleman  has  at  interest  $10,640  at  5#,  $37,500  at  6/c, 
and  $26,000  at  6y>$.  What  income  does  he  derive  from  these 
sums  per  annum  ? 

2.  Mrs.  Stone  has  24  government  bonds  of  $1000  each,  bear- 
ing 4$  interest.     What  is  her  income  per  quarter  ? 

3.  $1850  yields  $55  %  int.  in  6  months.     What  rate  is  paid  ? 

4.  A  principal  of  $500  was  increased  by  $35  interest  per  year. 
W^hat  was  the  rate  per  cent.  ?  In  how  many  years  will  the  prin- 
cipal be  doubled  by  simple  interest  ? 

5.  A  sum  of  money  was  invested  at  3%$  interest.  After  3 
years  4  months  the  amount  was  $10,887.50.     Find  the  principal. 

6.  What  is  the  principal,  if  after  %  year  the  amount  is  $413, 
the  rate  being  6%^  per  annum  ? 

7.  Mr.  A.  had  a  mortgage  on  his  house,  and  paid  6y4$  inter- 
est.    In  4  years  the  amount  was  $5262.50.     What  was  the  debt  ? 

8.  A  lady  inherits  $6480,  and  desires  to  derive  $32.40  per 
month  from  it.     At  what  rate  must  it  be  invested  ? 

9.  In  what  time  will  $462.50  produce  $37  interest  at  4^  ? 

10.  In  what  time  will  $723  produce  $60 y4  interest  at  5^? 

11.  What  sum  will  yield  $35  interest  at  7$  in  1  yr.  4  mo.? 

12.  What  principal  will  pay  $21.59  interest  at  5f0  in  8y2 
months  ? 

13.  Mr.  A.  borrowed  a  sum  of  money  at  5y2$,  and  after  V/j 
yr.  paid  the  amount  $4330.     What  was  the  principal  ? 


INTEEEST.  329 

14.  What  principal  gives  183 ?/8  interest  per  month  at  5$  ? 

15.  A  gentleman  draws  $2940  per  year ;  4/5  of  his  money  bears 
4%  Vs  5$.     Find  the  principal. 

16.  A  principal  of  $6000  has  by  simple  interest  grown  to 
$8820  in  a  number  of  years  ;  y3  of  the  time  it  brought  3  fo,  l/i  of 
the  time  5$,  and  the  remainder  of  the  time  4$  per  annum.  How 
long  did  the  principal  stand  ? 

17.  The  discount  at  6$  on  a  note  due  Nov.  1,  and  sold  on 
May  1,  was  $13  y2.     What  was  the  face  of  the  note  ? 

18.  I  had  a  note  for  $250,  due  in  21/2  months,  and  sold  it  at 
a  discount  of  1$  per  month.     How  much  did  I  get  for  it  ? 

19.  If  a  person  wishes  to  get  the  same  interest  for  $1200  in  4 
yr.  which  he  receives  for  $1000  at  4$  in  6  years,  what  rate  must 
he  charge  ? 

20.  In  how  many  years  will  $820  at  6yg#  produce  $278. 82  y, 
interest  ? 

21.  In  how  many  years  will  a  principal  of  $5000  grow  to  be 
$8000,  if  put  out  at  the  rate  of  6$  ? 

22.  The  sum  of  $3360  is  at  4x/2^  interest,  and  has  thus  far 
yielded  $1058.40  interest.  How  long  has  the  principal  been 
standing  ? 

23.  What  is  the  rate  of  discount  if  a  note  of  $300,  due  June 
20th,  is  sold  April  20th  for  $297  y8  ? 

24.  What  is  the  face  of  a  note,  sold  1  month  before  it  fell  due 
at  9$  discount,  the  discount  amounting  to  $15%  ? 

25.  Mr.  M.  buys  a  house  by  paying  13/20  of  the  purchase  money, 
and  securing  the  remainder,  $3500,  by  a  mortgage  at  4y2$.  At 
the  end  of  1  year  he  pays  the  remainder.  How  much  does  the 
house  cost  him  ? 

26.  Three  principals,  of  which  the  first,  $600,  has  been  at  6$ 
interest  for  2  i/2  years ;  the  second,  $425,  at  4$  for  3  years ;  and 
the  third,  $550,  for  2  years,  together  yield  $190.50  interest.  At 
what  rate  was  the  third  principal  invested  ? 


330  STANDARD  ARITHMETIC. 

27.  What  principal  will  bear  as  much  interest  in  6  years  at 
5$  as  $840  in  8  years  at  4$  ? 

28.  What  interest  will  you  get  on  $960  in  7  years,  if  $280  in 
5  years  yields  $63  interest  at  the  same  rate  ? 

29.  If  I  borrow  $480  at  5$  per  annum,  and  the  interest  for 
the  first  year  is  deducted  at  once,  what  per  cent,  do  I  really  pay  ? 

30.  $400  were  at  5^  interest  for  3y2  years,  and  $350  at  4$ 
for  a  longer  time.  The  principals  with  interest,  when  collected, 
amounted  to  $890.    How  long  was  the  second  principal  standing  ? 

31.  A  gentleman  borrows  $60,  and  promises  to  pay  $70  in  3 
months.     What  rate  per  cent,  will  he  pay  ? 

32.  A  guardian  put  his  ward's  money,  $17,500,  at  interest : 
%  of  the  money  at  4$,  and  the  remainder  at  6y8#.  How  much 
could  he  lay  by  annually  for  the  future  benefit  of  his  ward,  after 
deducting  $650  per  year  for  expenses  ? 

33.  For  what  time  does  Mr.  B.  pay  interest,  if  he  pays  $84  on 
$5600  at  the  rate  of  4%$  ? 

34.  Mr.  Frank  buys  a  house  and  pays  %  of  the  price  in  cash. 
The  remainder,  $5400,  is  secured  by  a  mortgage,  and  paid  in  2 
years  together  with  7y2$  interest.  How  much  was  paid,  in- 
cluding the  interest  ? 

35.  A  man  was  asked  what  money  he  had  at  interest.  He  said : 
One  half  of  it  yields  47§  A  the  other  half  5$,  and  I  receive  on 
the  whole  $114  a  month.     How  much  had  he  at  interest  ? 

36.  Mr.  Conklin  having  bought  a  piano  for  $350,  he  rented  it 
at  once  for  fifteen  months  at  $4y2,  payable  monthly.  Then  he 
sold  it  for  $325.  How  much  did  he  gain  by  the  transaction, 
taking  into  account  the  interest  on  the  cost  and  interest  on  pay- 
ments received  for  rent  ? 

37.  I  paid  $600  per  year  rent  for  a  factory  which  I  afterward 
bought  for  $12,000.  I  gave  $5000  cash  (which  was  worth  6$)  and 
a  4$  mortgage  for  the  balance.     How  much  per  year  did  I  save  ? 


INTEREST.  331 

Compound    Interest. 

333.  Compound  interest  is  interest  on  interest. 

Payment  of  compound  interest,  or  interest  on  interest,  can  not  be  enforced 
by  law. 

334.  Interest  is  usually  compounded  at  specified  intervals,  as 
annually,  semi-annually,  etc.,  by  adding  interest  to  principal,  and 
computing  interest  on  the  amount. 

Example. — l.  Find  the  compound  interest  of  $500,  at  6$,  for 
3  yr.  5  mo. 

$500.00       Principal.  $530 

30.00       Interest  1st  year.  - 

-$E20W  Amount.  _$3U»  Int.  2d  yea, 

31.80       Interest  2d  year. 
$561.80       Amount.  $561.80 

33.708     Interest  3d  year. 


$33.7080  Int.  3d  year. 


$595,508     Amount. 

14.8877  Interest  5  mo. 
"  **ia  o*~h    *  j.   o  *  $5.95508  Int.  2  mo. 

$610.39o7  Amount  3  yr.  5  mo.  -^ttttt^  T  *  , 

Laa  f\  •   •      i       •      •      i  $11.91016  Int.  4  mo. 

$500  Original  principal.  2  97754    "    l    " 

$110.3957  Compound  int.  3  yr.  5  mo.  $14.8877    Int.  5  mo. 

2.  What  is  the  compound  interest  on  $7325  for  2  yr.  2  mo. 

at  7  $  ?     (Carry  the  work  to  four  decimal  places.) 

3.  Find   the   compound    interest   on    $3333,    at   3y3#   semi- 
annually, for  1  yr.  7  mo. 

4.  What  amount  was  due  March  25,  1886,  on  $1512,  borrowed 
Jan.  25,  1885,  with  compound  interest  at  1 1/2  $  quarterly  ? 

5.  What  is  the  amount  of  $4615,  at  compound  interest,  for 
2  yr.  5  mo.  at  8  $  ? 

6.  Find  the  amount  of   $3500,   at  compound  interest,   from 
Oct.  29,  1884,  to  Nov.  15,  1885,  at  2  ^  quarterly. 

7.  How  much  greater,  at  compound  than  at  simple  interest, 
would  be  the  amount  of  $1568  in  3  yr.  8  mo.  at  6  </0  ? 

8.  Find  the  amount  due  Sept.  18,  1876,  on  $450,  loaned  Sept* 
18,  1873.     Interest  compounded  annually  at  4%$. 


332 


STANDARD  ARITHMETIC. 


The  use  of  the  following  table  will  greatly  shorten  calcula- 
tions in  compound  interest  : 

TABLE  SHOWING  THE  AMOUNT   OF    $1    AT    DIFFERENT  RATES,  COMPOUND 
INTEREST,   FROM  1  TO  15  YEARS. 


Yrs. 

3  per  cent. 

3J4  per  cent. 

4  per  cent. 

4%  per  cent. 

5  per  cent. 

6  per  cent. 

Yrs. 

1 

1.030000 

1.035000 

1*040000 

1.045000 

1.050000 

1.060000 

1 

2 

1.060900 

1.071225 

1.081600 

1.092025 

1.102500 

1.123600 

2 

3 

1.092727 

1.108718 

1.124864 

1.141166 

1.157625 

1.191016 

3 

4 

1.125509 

1.147523 

1.169859 

1.192519 

1.215506 

1.262477 

4 

5 

1.159274 

1.187686 

1.216653 

1.246182 

1.276282 

1.338226 

5 

6 

1.194052 

1.229255 

1.265319 

1.302260 

1.340096 

1.418519 

6 

7 

1.229874 

1.272279 

1.315932 

1.360862 

1.407100 

1.503630 

7 

8 

1.266770 

1.316809 

1.368569 

1.422101 

1.477455 

1.593848 

8 

9 

1.304773 

1.362897 

1.423312 

1.486095 

1.551328 

1.689479 

9 

10 

1.343916 

1.410599 

1.480244 

1.552969 

1.628895 

1.790848 

10 

11 

1.384234 

1.459970 

1.539454 

1.C22853 

1.710339 

1.898299 

11 

12 

1.425761 

1.511069 

1.601032 

1.695881 

1.795856 

2.012196 

12 

13 

1.468534 

1.563956 

1.665074 

1.772196 

1.885649 

2.132928 

13 

14 

1.512590 

1.618695 

1.731676 

1.851945 

1.979932 

2.260904 

14 

15 

1.557967 

1.675349 

1.800944 

1.935282 

2.078928 

2.396558 

15 

9.  Find  the  compound  interest  of  $1250,  at  6  #,  for  3  years. 

Analysis. — According  to  the  foregoing  table,  each  dollar  at  6  %  compound  in- 
terest for  3  years  would  amount  to  $1.191016,  and  the  amount  of  $1250  for  the 
same  time  and  at  the  same  rate  would  be  1250  times  as  much  =  $1488.77.  The 
principal,  $1250,  being  subtracted  from  $1488.77,  the  remainder  is  the  compound 
interest  =  $238.77  Am. 

Note. — Except  for  large  sums  of  money,  it  is  not  necessary  to  use  more  than 
four  decimal  places,  as  in  the  following  problems. 

10.  Find  the  amount  of  $750,  at  5  $,  compound  interest,  from 
March  10,  1883,  to  Sept.  10,  1886. 

11.  $325  was  placed  at  compound  interest  at  4  $  semi-annually, 
on  Jan.  1,  1882.     What  was  due  Jan.  1,  1886  ? 

12.  What  is  the  compound  interest  of  $650,  at  3  %  $  annually, 
for  1  y8  years  ? 

13.  Find  the  amount  paid  Oct.  31,  1886,  for  $1225,  borrowed 
Dec.  10,  1881.     Interest  compounded  at  3  $  semi-annually. 


CHAPTER   XVI. 

EQUATION    OF   PAYMENTS. 

ORAL    EXERCISES. 

In  what  time  will  the  use  of  $1  balance  the  use  of 

1.  $3  for  1  month  ?         4.  $3  for  2  mo.?         7.  $8  for  5  mo.? 

2.  $7  for  1  month  ?         5.  $7  for  2  mo.  ?         8.  $12  for  8  mo.  ? 

3.  $9  for  1  month  ?         6.  $9  for  2  mo.  ?         9.  $15  for  4  mo.  ? 

In  what  time  will  the  use  of  $1  balance  the  use  of 

10.  $8  for  4  mo.  +  $5  for  5  mo.  +  $9  for    2  mo.  ? 

11.  $7  for  2  mo.  +  $4  for  6  mo.  +  $5  for  10  mo.? 

12.  $8  for  2  mo.  +  $9  for  3  mo.  -f  $8  for     5  mo.  ? 

13.  How  long  should  $2  be  kept  in  use  to  balance  the  use  of 
$1  for  4  months  ? 

How  long : 

14.  $3  to  bal.  $2  for    2  mo.  ?  17.  $300  to  bal.  $200  for  6  yr.  ? 

15.  $5  to  bal.  $3  for  10  mo.  ?  18.  $400  to  bal.  $300  for  12  yr.  ? 

16.  $7  to  bal.  $9  for  14  mo.  ?  19.  $5000  to  bal.  $250  for  20  yr.  ? 

20.  How  long  should  $8  be  kept  at  interest  to  balance  the  in- 
terest of  $6  for  2  months  +  the  interest  of  $2  for  12  mo.  ? 

How  long : 

21.  $7  to  bal.  the  interest  of    $3  for  14  mo.  +   $4  for    3  %  mo.  ? 

22.  $15  to  bal.  the  interest  of    $9  for    5  mo.  +    $6  for    7 %  mo.  ? 

23.  $38  to  bal.  the  interest  of  $20  for    9  mo.  +  $18  for  11%  mo.  ? 

24.  How  long  a  time  may  $600  be  kept  in  use  to  balance  the 
use  of  $400  for  3  years  +  $200  for  9  years  ? 
15 


334  STANDARD  ARITHMETIC 

Definitions. 

335.  Equation  of  Payments  is  the  process  of  determining 
the  date  at  which  two  or  more  debts  due  at  different  times  may- 
be paid  in  one  sum,  without  loss  of.  interest  to  either  party. 

336.  The  Term  of  Credit  is  the  time  allowed  for  the  pay- 
ment of  a  note  or  account. 

337.  A  debt  is  said  to  mature  at  the  expiration  of  the  term 
of  credit. 

338.  The  Equated  or  Average  Time  of  payment  is  the  date 
at  which  several  items  of  debt  due  at  different  times  may  be  equi- 
tably paid  in  one  sum. 

WRITTEN     EXERCISES. 

Example. — Find  the  average  term  of  credit  of  $200  due  in  3 
mo.,  $450  due  in  4  mo.,  $500  due  in  4y8  mo.,  $350  due  in  5  mo. 

The  use  of   $200  for   3   mo.    is  $200  x  3  mo.  =    600  mo. 

equivalent  to  the  use  of   $1  for  200  $45Q  x  4  mo   =  ]800  mo< 

times  3  mo.  =  600  mo.  %m  ^  ^  mQ   =  jjj^j  mQ 

The  pupil  may  make  a  similar  ex-  $350  x  g  mo   =  mo  mo> 

planation  for  eaeh  item.  ^—                               —  ^ 

The    use    of    $1500  in  parts,   as  Am%  4*/l4  mo.  =  4  mo,  8  d. 

specified  in  the  example,  that  is,  $200 

for  3  mo.,  $450  for  4  mo.,  etc.,  is  thus  found  to  be  equivalent  to  the  use  of  one 
dollar  for  6400  mo.  But  the  use  of  $1  for  6400  mo.  is  equivalent  to  the  use  of 
$1500  for  Visoo  of  6400  mo.  =44/is  mo.,  or  4  mo.  8  d.  Am. 

Hence,  to  find  the  average  term  of  credit  for  several  sums  of 
money,  due  at  different  times,  by  the 

Method  of   Products. 

339.  Utile,— Multiply  each  item  of  the  debt  by  its  term  of 
credit,  and  divide  the  sum  of  the  products  by  the  sum  of  the 
items ;  the  quotient  will  be  the  average  term  of  credit. 

Notes. — 1.  In  computing  terms  of  credit,  it  is  customary  to  reject  the  cents  in 
any  item  if  less  than  50 ;  and,  if  50  or  greater,  to  reckon  them  as  one  dollar. 

2.  Less  than  1/2  day  in  a  result  is  rejected;  a  1/2  day  or  greater  fraction  is 
counted  as  1  day. 


EQ  UA  TION  OF  PA  YMENTS.  335 

The    Interest   Method. 

34-0.  The  time  in  which  the  sum  of  several  items  of  indebted- 
ness would  become  justly  due  is  the  time  in  which  the  use  of  the 
sum  would  balance  the  use  of  the  items  for  the  several  terms 
allowed  for  their  payment.     Thus,  in  the  foregoing  problem, 

The  use  of  $200  for  3      mo.  at  6f0  would  be  worth    $3.00 

"       "      $450    "    4         "  u  "  "  $9.00 

"       "      $500    "    4V2  "  <fc  "  u        $11-25 

"       "      $350     "    5         u  "  "  "  $8.75 

$1500  $32.00 

Thus  we  find  that  the  use  of  the  several  items  as  allowed  by  agreement  is  worth 
$32.  The  question  then  is,  How  many  months'  use  of  $1500  would  be  worth  as 
much  ?     Hence  the 

Analysis. — The  use  of  $1500  for  1  mo.  at  6%  per  annum  is  worth  $7.50,  and, 
that  its  use  may  be  worth  $32,  it  must  remain  at  interest  as  many  times  1  mo.  as 
there  are  times  $7.50  in  $32  =  4  4/15  times ;  4  4/16  times  1  mo.  =  4  mo.  8  d.  Am. 

341.  Any  per  cent,  maybe  used  in  finding  equated  time  by 
the  method  of  interest.  In  solving  the  foregoing  example,  for 
instance,  we  may  use  12$  per  ann.  =  1$  per  month,  as  follows  : 

The  interest  of  $200  for  3      mo 

"  "         $450    "    4         " 

"  "         $500    "    4V2  " 

"         $350    "    5         " 

$1500 

$64 -$15  =  4*/!  5.     *.*/« 

The  use  of  the  several  items  for  the  terms  of  credit  allowed  having  been  found, 
as  above,  to  be  worth  $64,  we  divide  $64  by  the  interest  of  $1500  for  1  month  at 
l?c  to  find  the  term  of  credit  for  $1500.  The  interest  of  $1500  for  1  month  at 
1  %  per  month  =  $15.  $64  -+-  $15  =  4  4/15.  Ans.  44/15  mo.  =  4  mo.  8  d.  Hence 
the  following : 

342.  Rule.—l.  Find  the  interest  of  each  item  of  debt  for  its 
term  of  credit  at  any  assumed  rate  per  cent. 

2.  Divide  the  sum  of  the  interests  by  the  interest  of  the  whole 
debt  for  any  unit  of  time  (day,  month,  or  year),  and  the  quotient 
will  be  the  average  term  of  credit  in  the  denomination  of  the 
unit  selected. 


It     lfo 

per  mo 

.  =    $6.00 

14 

u 

=  $18.00 

H 

a 

41 

=  $22.50 

=  $17.50 

$64.00 

mo.  = 

:  4  mo. 

8  d. 

336  STANDARD  ARITHMETIC. 

SLATE     EXERCISES. 

1.  May  1,  1889,  I  purchased  property  for  $8500,  paid  cash 
$1500,  and  gave  notes,  one  for  $3000,  payable  in  2  years,  and 
another  for  $4000,  payable  in  4  years.  Find  the  average  term 
of  credit  on  the  notes. 

2.  Sept.  1,  1881,  I  bought  goods,  as  follows  :  $200  on  2  mo. 
time,  $400  on  3  mo.,  and  $450  on  4  mo.  What  was  the  average 
term  of  credit,  and  the  average  date  of  maturity  ? 

Ans.  The  average  term  of  credit  was  3  mo.  8  d.,  which,  being  added  to  Sept.  1, 
brought  the  average  date  of  maturity  on  Dec.  9,  1881. 

3.  Jan.  15,  I  bought  a  bill  of  goods  amounting  to  $900,  $275 
of  which  was  on  30  days'  credit,  $300  on  60  days,  and  $325  on 
90  days.     What  was  the  equated  time  of  payment  ? 

4.  James  Hudson,  June  12,  owes  $317.75  due  in  4  mo.,  $216.38 
due  in  5  mo.,  and  $170  due  in  6  mo.  Find  the  equated  time 
of  payment  and  date  of  maturity. 

5.  William  Owens  bought  a  farm  of  320  acres  at  $68  per  acre, 
V«  payable  in  cash,  %  in  1  year,  y3  in  3  years,  and  the  remainder 
in  5  years.     What  was  the  average  term  of  credit  ? 

6.  Find  the  average  term  of  credit : 

$189.50  on  90  d.  $560.00  on  90  d.  $120.00  on  90  d. 

$150.00  <k  60  d.  $83.50   u  45  d.  $80.00  u  30  d. 

$70.00  "  90  d.  $15.00   "  60  d.  $480.00  "  90  d. 

7.  Mrs.  Handy  bought  a  city  lot,  May  1, 1879,  paying  $60  cash, 
$120  in  10  mo.,  $150  in  15  mo.,  and  $200  in  20  mo.  What  was 
the  average  term  of  credit  on  deferred  payments  ? 

8.  On  Feb.  1,  1880,  Mrs.  Handy  paid  the  whole  amount  due ; 
what  discount  was  allowed  her  at  6  fo  (bank  discount)  ? 

9.  Bought  5000  bu.  of  coal  at  13^  a  bu.,  payable  in  30  days. 
But  I  paid  $300  after  10  days  ;  what  term  of  credit  was  I  en- 
titled to  on  the  balance  ? 

10.  A  debt  of  $2400  was  contracted  March  6,  1876,  payable  in 
8  mo.,  but  $400  was  paid  in  2  mo.,  $600  in  5  mo.,  $800  in  7  mo. 
What  was  the  equitable  time  for  paying  the  balance  ? 


I1/* 

X 

1 

mo. 

= 

%u 

moo 

v« 

X 

3 

mo. 

= 

3U 

mo. 

V. 

X 

4 

mo. 

— 

2/3 

mo. 

V. 

X 

5 

mo. 

= 

5/3 

mo. 

EQUATION  OF  PAYMENTS.  337 

11.  A  stock  of  groceries  was  purchased  Jan.  1,  1886,  the  pur- 
chase price  payable  as  follows  :  %  in  1  mo.,  */4  in  3  mo.,  */,  in 
4  mo.,  y3  in  5  mo.  When  may  the  whole  amount  be  equitably 
paid  in  one  sum  ? 

Inasmuch  as  the  relation  of  each  item 
of  indebtedness  to  the  whole  amount,  and 
its  term  of  credit,  are  the  only  conditions 
necessary  to  be  known,  $1  may  be  assumed 
as  the  purchase  price,  and  the  operation  in 
many  cases   be   performed   orally,  as  here  *  *■  =  "  l»  m0* 

indicated. 

12.  John  I)oe  sells  to  Richard  Roe  goods  to  the  amount  of 
$3600  ;  y4  on  2  mo.  credit,  y3  on  3  mo.,  and  the  balance  on  4 
mo.     What  is  the  average  term  of  credit  ? 

13.  Nov.  1,  1881,  I  sold  a  horse  and  carriage  for  $650,  y4  pay- 
able in  3  mo.,  y4  in  4  mo.,  and  %  in  6  mo.  Find  the  equitable 
date  for  the  payment  of  the  whole  sum. 

14.  Jan.  12,  1880,  Thomas  Kline  sold  a  farm  for  $2890,  payable 
y4  in  cash,  y3  in  1  year,  and  the  balance  in  2  years  (no  interest). 
When  should  interest  begin  on  the  deferred  payments  ? 

15.  A  book  dealer  bought  a  stock  of  books  and  stationery  for 
$2400  on  4  mo.  time,  but  in  one  month  he  paid  $600,  and  in  2 
mo.  $800.     In  what  time  might  he  equitably  pay  the  balance  ? 

In  making  the  payments  before  ^qq   x   g  _  4g00 

due  he  lost  the  use  of  $600  for  3  mo.,  ftnn        9  _  1  fiftft 

and  of  $800  for  2  mo.,  which,  by  the  **UU  X  Z  ~  ib  U 

process    just    given,   we  find  to    be  1400  3400 

equivalent  to  the  use  of  $1  for  3400  2400  —  1400  =  1000 

mo.     He  was  therefore  entitled  to  keep  1/1000  of  3400  =  3  2/5     . 

the  remainder  of    the  debt    ($1000)        4  mo#  +  g»/    mQ<  =  ft /    mo#  Angm 
beyond  the  stipulated  time,  long  enough 

to  balance  the  loss.     The  use  of  $1  for  3400  mo.  —  the  use  of  $1000  for  the  V1000 
of  3400  mo.  =  3  2/5  months.     4  mo.  +  3  2/5  mo.  =  72/5  mo.  =  V  mo.  12  d.  Am. 

16.  Austin  &  Co.,  Oct.  12,  1880,  bought  a  bill  of  goods  to  the 
amount  of  $2480,  on  a  credit  of  4  mo.,  but  paid  $700  on  Nov.  9, 
and  $850  on  ISTov.  30.  Find  the  equitable  date  for  paying  the 
balance. 


338  STANDARD  ARITHMETIC. 

When  the  terms  of  credit  begin  and  mature  at  different  dates. 

Example. — Find  the  equitable  date  of  a  note  which  may  be 
given  in  the  settlement  of  the  following  account : 

Levi  Little 

to  Nelson  New  Dr. 

1881.  March  10,  to  rndss.  on  2  mo.  credit  $800 

"  "       19,  "  3  u  $620 

"       April      8,  "  4  "  $420 

%u      May      18,  "  3  "  $560 

Taking  the  dates  of  maturity  instead  of  the  dates  of  purchase 
as  given  in  the  account,  and  finding  the  number  of  days  from  the 
earliest  date  to  each  of  the  succeeding  ones,  we  obtain  the  average 
date  of  maturity  as  follows  : 


Due  dates. 

Items. 

Days.              Products. 

May    10, 

$800 

X          0     =                  0 

June    19, 

$620 

x       40     =       24,800 

Aug.      8, 

$420 

x        90     =       37,800 

"       18, 

$560 

x      100     =       56,000 

~$2400 

118,600 

-J-  2400  = 

495/i2- 

May  10  +  49  d.  =  June  28  Ans. 

118,600 

Or,  taking  the  difference  of  time  between  the  latest  and  each 
preceding  date  of  maturity,  and  computing  the  average  by  a  pro- 
cess similar  to  the  foregoing  one,  we  may  determine  the  equated 
time  of  payment  by  counting  backward  instead  of  forward  : 


Dates. 

Items. 

Days. 

Products. 

May     10, 

$800 

X 

100 

= 

80,000 

June    19, 

$620 

X 

60 

= 

37,200 

Aug.      8, 

$420 

X 

10 

= 

4,200 

"       18, 

$560 
$2400 

X 

0 

= 

0 
121,400 

1214  -=-  24  =  50 

7/i2d. 

Au 

g.  18- 

-  51  d.  =  Jui 

Thus  we  see  that  either  the  earliest  or  latest  date  of  maturity 
may  be  assumed  as  the  focal  date,  and  that  either  process  may  be 
used  to  test  the  accuracv  of  the  other. 


EQUATION  OF  PAYMENTS.  339 

EXAMPLES. 

Find  the  average  term  of   credit,  and  equated  time  of  pay- 
ment : 


1. 

Purchased  : 

2. 

Purchased  : 

M:ir. 

4, 

$450,    2  mo.  cr. 

Aug.  20, 

$500,    3  mo.  cr. 

May 

10, 

$316,    1       " 

Sept.     3, 

$380,    2       " 

July 

9, 

$420,    2       " 

Oct.    19, 

$295,    4       " 

Aug. 

1, 

$500,    3       " 

. "       30, 

$400,    1       " 

3.  A  young  man,  having  money  advanced  to  help  him  pay  his 
way  through  college,  received  : 

Sept.    1,  1878,     $75.  Feb.    15,  1880,       $86. 

Feb.    15,  1879,     $80.  Sept.  20,  1880,     $128. 

Aug.  31,  1879,     $95.  Aug.  30,  1881,    $175. 

What  was  the  equated  time  at  which  he  should  date  a  single  inter- 
est-bearing note  for  the  whole  sum  ? 

4.  Five  years  from  the  date  of  the  first  loan,  the  above-men- 
tioned note  was  paid,  with  interest  at  4  %  ;  what  was  the  amount  ? 

5.  What  is  the  average  time  at  which  the  following  bills  become 
due  ?  Feb.  10,  1882,  $400  on  2  mo.  credit ;  May  10,  1882,  $300 
on  4  mo.  credit ;  June  16,  1882,  $350  ;  Aug.  6,  1882,  $150. 

6.  Find  the  equitable  date  for  a  single  note  given  on  the  fol- 
lowing bills  for  merchandise:  June  1,  1885,  $20,  July  1,  $30,  Aug. 
1,  $30,  Sept.  1,  $20,  each  on  2  mo.  credit. 

7.  Bought  goods  of  Messrs.  Holt  &  Co.,  as  follows  :  Mar.  11, 
$35,  on  30  d.  credit ;  July  20,  $95,  on  2  mo.  credit ;  Sept.  8, 
$215,  on  3  mo.  credit.     What  was  the  average  term  of  credit  ? 

8.  A  credit  of  5  mo.  on  $400,  one  of  3  mo.  on  $900,  and  one 
of  7  mo.  on  $600,  are  equivalent  to  a  credit  on  how  many  dollars 
for  one  year  ? 

9.  Sold  Mr.  Long  the  following  goods  :  May  2,  two  dozen 
ulsters,  @  $18  each,  on  3  mo.  credit ;  June  21,  six  dozen  vests, 
@  $2.50  each,  on  2  mo.  credit ;  Aug.  1,  three  dozen  pique  pants, 
@  $4  each,  on  40  days'  credit.  Find  the  average  term  of  credit 
and  the  equated  time  of  payment. 


340 


STANDARD  ARITHMETIC. 


Debit  and  Credit  Accounts. 

l.  Find  the  equitable  balance  of  the  following  account : 
Dr.         John  Loch  in  account  with  Geo.  Putnam.         Cr. 


1880. 

1880. 

Oct.     2, 

To  mdse. 

$180 

Nov.  18, 

By  mdse.  (2  mo.) 

$150 

Nov.    8, 

"         (3  mo.) 

$120 

Dec.  24, 

"  cash, 

$200 

Dec.  16, 

"         (4  mo.) 

$240 

Explanation. — It  is  to  be  noticed  that  April  16,  1881  (4  mo.  after  Dec.  16),  was 
the  latest  date  for  the  complete  maturity  of  any  transaction  recorded  in  the  fore- 
going account.     We  therefore  look  into  it  to  see  how  it  stood  on  that  day. 

It*  the  value  of  the  goods  bought,  and  the  money  paid,  were  the  only  matters 
for  consideration,  the  sum  due  to  Putnam,  April  16,  would  have  been  $190.  But 
each  had  also  had  from  the  other  the  use  of  money  from  the  dates  when  it  became 
due,  or  the  days  of  payment,  to  the  time  of  settlement.  Hence,  we  find  how  many 
days'  use  of  $1  is  equivalent  to  the  advantage  which  each  one  thus  received  from 
the  other  from  and  after  each  date  to  April  16. 

Solution. 


Due. 
Oct.  2, 
Feb.  8, 
April  16 

Am't.         Days. 
$180    x     196 

$120    x      6V 
$240    x        0 
$540    x       ? 
$350 

Product.                               Due.      Am't.         Days.      Product. 
=   35,280                         Jan.  18,  $150    x      88   =    13,200 

=     8,040                          $200    x    113   =   22,600 

$350                        35,800 

=  43,320  no.  days'  int.  on  $1,  due  Putnam  by  Lock. 
35,800         u           u         $1,  which  Lock  had  paid. 

Leaving  $190  cash  bal.  and  7,520  bal.  of  days'  interest  on  $1  due  to  Putnam. 
7520  -=-  190  =  39 x  */,  9  d.     Practically,  40  days.  Ans. 

From  this  solution  we  find  that,  on  the  16th  day  of  April,  Lock  owed  Putnam 
not  only  $190,  but  also  an  intrrest  balance  equivalent  to  the  interest  of  $1  for  7520 
days,  which  was  equal  to  $1.25. 

Evidently  this  difference  could  have  been  adjusted  by  Lock  paying  Putnam 
$190  +  $1.25  =  $191.25.  This  was  the  cash  balance  due  April  16.  But  if  not  pre- 
pared to  pay  the  cash,  Lock  would  have  given  his  note  for  $190,  with  interest, 
dated  backward  40  days  from  April  16,  so  that  it  might  be  worth  $191.25  on  the 
day  it  was  given. 

If  the  balance  of  items  and  the  interest  balance  had  been  on  the  other  side  of 
the  account,  Putnam  would  have  been  the  debtor,  and  would  have  had  to  give  his 
note  dated  back  to  the  equated  time  of  maturity,  just  as  Lock  was  supposed  to  do. 

But  if  the  balance  of  items  had  been  on  one  side,  and  the  interest  balance  on 
the  other,  the  average  date  of  maturity  would  have  been  thrown  forward  instead  of 
backward,  and  the  note  would  have  been  dated  accordingly. 


EQUATION  OF  PAYMENTS. 


341 


SLATE    EXERCISES 


2.  When  did  a  note  given  in  settlement  of  the  following  ac- 
count begin  to  bear  interest : 


Dr. 


L.  R.  Clem, 


Cr. 


July  2,        To  mdse.  (3  mo.)      $580       Aug.  14,       By  cash, 


$450 


3.  When  did  interest  begin  on  the  following  account,  and  what 
was  due  on  settlement,  Jan.  1,  1882  : 


Dr. 


C.  L.  Hoosacfa 


Cr. 


1881. 

1881. 

June  17, 

To  mdse.  (2  mo.) 

$270 

June  30, 

By  mdse. 

$250 

Sept.  20, 

(3  mo.) 

$650 

Oct.  1, 

By  cash, 

$500 

Oct,  1, 

"        (1  mo.) 

$100 

Nov.  30, 

By  mdse. 

$150 

4.  Find  the  cash  balance  due  on  the  following  account  on  the 
latest  day  of  maturity  : 

Dr.  W.  M.  Davis.  Cr. 


1882. 

1882. 

Mar.  30, 

To  mdse.    (60  d.) 

$300 

Mar.  10, 

By  mdse. 

$180 

April  2, 

"          (90  d.) 

$700 

June  20, 

u 

$980 

July  16, 

"          (60  d.) 

$150 

July  27, 

By  draft, 

$290 

5.  Find  the  equated  time  for  the  payment  of  the  balance  due 
on  the  following  account : 

Cr. 


Dr. 


W.  T.  Dawes. 


1882. 

1882. 

Mar.  1, 

To  mdse.    (60  d.) 

$200 

Mar.  6, 

By  mdse. 

$200 

May  10, 

»         (60  d.) 

$900 

May  16, 

By  cash, 

$150 

June  20, 

(90  d.) 

$400 

June  26, 

t< 

$360 

July  30, 

"          (30  d.) 

$700 

July  1, 

(4 

$990 

Au«r.  14, 

(60  d.) 

$100 

Aug.  28, 

By  mdse. 

$240 

342  STANDARD  ARITHMETIC. 

Original  Problems. 

1.  Any  actual  business  transactions  that  may  with  propriety 
be  reported  to  the  class  can  be  made  a  subject  for  one  or  more 
original  problems. 

2.  Obtain  from  parents  or  friends  copies  of  notes,  the  names 
thereon  being  changed  ;  and  ask  the  class  to  compute  interest, 
amount,  and  proceeds  at  bank  at  current  rates. 

3.  Ascertain  about  what  it  costs  per  year  to  board  and  clothe 
a  school  boy  or  girl,  and  how  much  money  must  be  invested  at 
current  rates  to  produce  that  much  interest. 

4.  Suppose  that  you  buy  a  vacant  lot  for  a  given  sum,  and 
in  a  number  of  years  sell  it  for  less  than  you  gave  for  it.  Ask 
the  class  how  much  you  would  lose  by  the  transaction,  supposing 
the  use  of  your  money  to  be  worth  current  rates. 

5.  Construct  for  the  class  a  problem  in  annual,  compound, 
exact  interest,  etc. 

6.  Ask  what  sum  you  should  put  at  interest  that  in  a  given 
number  of  years  it  may  amount  to  enough  to  erect  a  public  library 
building  at  any  cost  you  think  desirable. 

7.  Find  the  rent  paid  for  some  particular  house,  and  what  the 
taxes  on  it  are,  and  ask  the  class  whether  it  would  be  profitable 
to  buy  it  at  the  price  asked,  not  forgetting  insurance  at  current 
rates  and  the  use  of  money. 

8.  Ask  which  is  to  be  preferred  by  the  creditor,  compound  or 
annual  interest,  and  how  much  one  would  be  worth  more  than 
the  other  on  any  given  sum  for  any  given  time  and  rate  per  cent. 

9.  Ask  the  yearly  interest  on  some  of  the  national,  State,  or 
city  loans  that  may  be  noticed  in  the  newspapers  from  time  to 
time.     On  the  debt  of  your  own  city. 

10.  Write  a  promissory  note,  indorse  three  or  four  partial  pay- 
ments, and  ask  the  amount  due,  etc. 

11.  Suppose  some  business  transactions  with  a  school-mate, 
which  require  an  equation  of  payments. 


CHAPTER   XVII. 

PROPORTION. 


ORAL     EXERCISES. 

1.  A  boy  buys  3  lemons  for  7^.  How  much,  at  the  same  rate, 
would  he  have  to  pay  for  6  lemons  ?     9,  12  lemons  ? 

Suggestion. — 6  lemons  are  how  many  times  8  lemons  ?  How  many  times  as 
much  will  they  cost? 

2.  If  3  dozen  of  eggs  are  worth  25^,  what  is  the  worth  of  9 
doz.?    15  doz.?     12  doz.? 

3.  Mr.  Jones  walks  17  miles  in  5  hours.  At  the  same  rate, 
how  many  miles  will  he  walk  in  15  h.  ?    30  h.  ?    40  h.  ? 

4.  Tf  6  oranges  can  be  had  for  21^,  what  will  2  cost  ? 

Suggestion. — What  part  of  6  oranges  is  2  oranges  ?  What  part  of  the  cost 
of  6  should  be  the  cost  of  2  ? 

5.  If  14  bl.  flour  cost  $35,  how  many  can  be  had  for  $105  ? 

6.  If  a  number  of  hands  can  plow  5  acres  in  6  hours,  how 
many  acres  can  they  plow  in  48  hours  ? 

7.  Sixteen  lb.  cost  36^.  At  the  same  rate,  what  will  12  lb. 
cost  ?. 

8.  A  lot,  32  ft.  wide,  costs  $500.  What  will  a  lot,  measuring 
96  ft.  in  width,  cost  at  the  same  rate  ? 

9.  A  courier  travels  on  an  average  156  miles  in  3  days  ?  How 
far  will  he  travel  in  12  days  ?     18  d.  ?     24  d.  ? 

10.  What  is  the  height  of  a  steeple,  that  casts  a  shadow  of 
300  feet,  if  at  the  same  time  a  staff,  2  feet  high,  casts  a  shadow 
of  3  feet  ? 


344  STANDARD  ARITHMETIC. 

SLATE     EXERCISES. 

1.  If  a  man  can  earn  $23  in  13  days,  how  much  can  he  earn, 
at  the  same  rate,  in  221  days  ? 

Solution. — We  know  that,  at  the  same  rate  of  wages  per  day, 

221  days' mugtbeagtimeg  13  days'  ag  ^  days  are  times  13  days, 
wages  wages  J  J 

But,  since  the  wages  of  13  days  are  $23,  we  may  as  well  have  written 

'      are  as  many  times    $23    as  221    daVS  are  times  13    da\7S. 

wages  *  J 

And  since  221  days  are  17  times  13  days,  the  wages  of    221  days  must  be  17 
times  $23  =  $391  Am. 

Inasmuch  as  the  words  printed  in  small  type  are  invariably  the  same,  whatever 
the  nature  of  the  question,  signs  may  be  substituted  for  them.     Thus, 

221  days'  _^_  $23  =  ^J  d        -   ig  d 

wages  "  J 

This  form  may  be  read  exactly  as  the  preceding  one,  though  it  is  more  common 
in  this  case  to  use  the  colon  ( : )  for  the  sign  of  division  (-i-)  and  a  double  colon  ( : : ) 
for  the  sign  of  equality.     Thus, 

221  days'       Q23       m  d  13   , 

wages  J  J 

Since  221  days  are  17  times  13  days,  the  wages  for  221  days  must  be  17  times 
the  wages  of  13  days.     17  times  $23  =  $391  Am. 

2.  A  man  who  travels  at  the  rate  of  258  miles  in  13  days  will 
travel  how  far  in  32 1/2  days  ? 

Miles  traveled  /0\     .    miles   . .   days    .    days 
in  32 Vi  days  \\J     '      258     '*    32%    '      13. 

Explanation. — The  man  will  travel  as  many  times  258  miles  in  32  Y2  days  as 
32  xj2  days  are  times  13  days.  Having  found,  from  the  second  pair  of  terms,  that 
32  x/2  days  are  ti1^  times  13  days,  and  hence  knowing  also  that  the  first  term  of 
the  first  pair  must  be  2 1/a  times  the  second  term,  we  make  it  so  by  multiplying 
258  by  2  */gJ  and  thus  obtain  the  required  answer. 

3.  A  lady  bought  15  yards  of  calico  for  $1.40.     At  the  same 

rate,  how  much  would  she  have  paid  for  42  yards  ? 

Statement. 
Cost  of  Cost  of  v„„qc,      v„^o 

42  yds.  15  yds.  Yards-      Yards-  g& 

?      :      $1.40     ::     42    :    15        Solution.— $1.40  X  =?  =  $3.92. 

15 


PROPORTION.  345 

4.  James  purchased  15  acres  of   farm-land  for  $72.      How 
much  did  William  have  to  pay  for  37  Yg  acres  at  that  rate  ? 

5.  A  locomotive  runs  18  miles  in  30  min.     How  many  miles 
does  it  run  in  50,  65,  72,  81  min.  ? 

6.  A  ship  sailed  47  Yi  miles  in  5  hours.     How  long  will  she 
be  in  sailing  180  miles  ? 

7.  If  7  men  eat  10  Yg  loaves  of  bread  a  week,  how  many  will 
25,  67,  39  men  eat  at  the  same  rate  in  the  same  time  ? 

8.  What  is  the  height  of  a  tower  which  casts  a  shadow  of  210 
feet,  when  a  pole,  15  feet  high,  casts  a  shadow  oi;  18  feet  ? 


Inverse  Proportion. 

Example. — l.  A  house  was  painted  by  8  men  in  6  days.  How 
many  men  would  have  been  required  to  do  the  same  work  in  12 
days  ? 

Note. — The  following  erroneous  statement  is  likely  to  be  made : 

The  number  of  men  )  is  as  C  the  number  of  men  re-  ) 
required  to  do  the  v  many  -j     quired  to  do  the  same  >  as  12  days  is  times  6  days, 
work  in  12  days     )  times  (     work  in  6  days  ) 

When  solved,  it  would  lead  us  to  the  conclusion  that  it  would  take  16  men  12  days 
to  do  a  work  which  requires  only  8  men  6  days,  which  is  absurd.  If  we  allow 
double  the  time  for  any  work,  we  do  not  need  twice  as  many  men  to  do  it,  but  only 
half  as  many. 

The  statement  then  should  be : 

The  number  of  men  )  (  the  number  of  men  ) 

required  to  do  the  [•  paJ"  5  \     required  to  do  the  [  as  6  d.  is  of  12  d. 
work  in  1 2  days      )  (     work  in  6  days       ) 

Men.       Men.  Days.       Days. 

Or,  using  the  shorter  form :      ?      :      8       ::       6      :      12. 

We  reason  that,  since  6  days  is  one  half  of  12  days,  the  number  of  men  re- 
quired to  do  the  work  in  12  days  is  only  one  half  as  many  as  would  be  required  to 
do  it  in  6  days.     Ans.,  4. 

In  all  the  problems  preceding  this,  more  required  more  ;  that 
is,  more  goods  required  more  money,  more  worh  required  more  men, 
etc.,  etc.  But,  as  we  see  in  this  example,  there  are  problems  in 
which  more  requires  less  and  less  requires  more. 


340  STANDARD  ARITHMETIC. 

ORAL     EXAMPLES. 

2.  If  a  man  can  perform  a  journey  in  6  days,  traveling  12 
hours  a  day,  how  many  days  will  be  required  if  he  travels  only  6 
hours  a  day  ?    4  hours  ?    3  hours  ? 

3.  The  owner  of  a  livery  stable  sends  16  horses  to  pasture  for 
8  days.  How  many  could  he  send  for  the  same  money  for  16 
days  ?    4  days  ? 

4.  Six  masons  can  perform  a  certain  work  in  24  days.  How 
long  will  it  take  12  men  ?     18  men  ? 

5.  Six  horses  consume  a  certain  quantity  of  oats  in  12  days. 
How  long  will  it  feed  36  horses  ? 

6.  Fifteen  farm  hands  will  mow  the  harvest  of  a  farm  in  8 
days.     How  many  will  do  it  in  2  days  ?    In  24  days  ? 


SLATE     EXERCISES. 

7.  Fred  has  to  walk  250  steps  from  his  house  to  school,  each 
of  his  steps  measuring  18  in.  His  brother's  steps  measure  27  in. 
each.     How  many  steps  does  his  brother  take  ? 

8.  A  messenger  who  traveled  12  miles  an  hour  reached  his 
point  of  destination  in  3  hours.  How  long  would  it  have  taken 
him  had  he  traveled  only  8  miles  per  hour  ?     6  miles  ?    4  miles  ? 

9.  If  he  had  traveled  4%  miles  an  hour,  how  many  hours 
would  it  have  taken  him. 

10.  Thirty  sailors  can  subsist  on  their  provisions  4  months. 
At  the  same  rate,  how  long  will  the  same  provisions  last  20  sailors  ? 

11.  Suppose  the  same  provisions  would  last  30  men  4y2  months. 
Find  how  many  months  and  days  they  would  last  20  men. 

12.  A  carter  agrees  to  transport  7%  cwt.  6  miles  for  a  certain 
sum.     How  far  will  he  carry  9  cwt.  for  the  same  money  ? 

13.  A  bridge  was  built  by  15  workmen  in  4  weeks  and  4  days. 
How  many  would  have  built  it  8  days  sooner  ? 

14.  A  certain  number  of  trees  was  felled  by  28  men  in  4  weeks 
and  3  days.     How  many  could  have  done  it  in  12  days  ? 


PROPORTION.  347 

Ratio  and  Proportion. 

343.  A  comparison  of  two  numbers  is  made  by  showing  how 
many  times,  or  what  part/ one  number  is  of  another. 

344-.  In  any  comparison  of  numbers  there  must  be  at  least 
hvo  numbers,  or  quantities,  compared. 

345.  Two  numbars  or  quantities  thus  compared  are  together 
called  a  couplet.  The  first  is  called  the  antecedent  (the  one  going 
before)  ;  the  second,  the  consequent  (the  one  coming  after). 

346.  The  antecedent  and  consequent  are  called  the  terms  of 
the  couplet. 

347.  The  consequent  is  the  standard  of  comparison. 

348.  The  relation  of  two  numbers,  that  is,  the  quotient  ob- 
tained by  dividing  the  antecedent  by  the  consequent,  is  called  the 
ratio  of  those  numbers. 

349.  Proportion  is  an  equality  of  ratios. 
For  instance,  the  following  is  a  proportion  : 

The  cost  of  9  yd.   m  as  many  the  cost  of  3  yd.        yd.     are     yd. 
$1*0  time8  12  780  a8     9     time8    3 

The  arithmetical  form  for  the  statement  of  which  is  : 

Cost.  Cost.  Quantity.        Quantity. 

37  V0   :   l&ftf  ::   9  yd.    :   3  yd. 

350.  The  double  colon   ::   is  the  special  sign  of  a  proportion. 

Note. — The  colon  ( : )  is  a  sign  of  division,  the  line  between  the  dots  being 
omitted.     The  sign  of  equality  (  =  )  is  often  used  instead  of  the  double  colon  ( : : ). 
Thus  the  statement  of  a  proportion  frequently  appears  in  this  form : 
Cost.  Cost.  Quantity.        Quantity. 

37V*0  f  1*70  =  9  yd.  -  3  yd. 

351.  A  proportion  must  contain  at  least  two  couplets.  The 
first  and  second  terms  make  the  first  couplet;  the  third  and 
fourth  terms,  the  second  couplet. 

352.  The  first  and  fourth  terms  of  a  proportion  are  called 
the  extremes;  the  second  and  third  are  called  the  means. 


348 


STANDARD  ARITHMETIC. 


EXERCISES    IN     FINDING     RATIOS 

Find  the  ratios  of  the  following  couplets  : 

The  consequent  being  the  standard  of  comparison,  the  question  to  be  answered,  in 
each  case,  is,  How  does  the  antecedent  compare  with  the  consequent  ?  How  many 
times,  or,  What  part  of,  the  consequent  is  the  antecedent  ? 


1.  18  :    9  = 

4.  95  :  19  = 

7.  13  :    4  = 

10.  42 

8  == 

2.  36  :  18  = 

5.  18  :  12  = 

8.  16  :  15  = 

11.  48 

5  = 

3.  72  :  12  = 

6.  85  :  17  = 

9.  51  :    3  = 

12.  88 

9    S3 

Note. — The  quotient,  arising  from  dividing  one  number  by  another,  may  be 
expressed  in  the  form  of  a  fraction,  thus,  15  -s-  3  =  15/3,  which,  being  narrowed 
down  to  lowest  terms,  is  equal  to  5.  In  writing  out  the  foregoing  exercises,  the 
pupil  may  therefore  adopt  the  following  form:  15  -f-  3  =  15/3  =  5.  He  should 
recollect  that  15-4-3,  15  :  3,  and  15/3,  indicate  the  same  thing,  viz.,  that  15  is  to 
be  divided  by  3. 

Questions. — When  we  compare  a  greater  number  with  a  less, 
is  the  ratio  greater  or  less  than  a  unit  ? — Can  a  ratio  be  expressed 
by  a  mixed  number  ?  By  a  fraction  ? — Why  is  the  ratio  of  32  :  8 
(read,  3^  to  8)  not  greater  than  that  of  8  :  2  ? — Give  other  couplets 
having  the  same  ratio  as  32  :  8.  As  15  :  3,  etc. — If  you  double 
both  numbers  does  the  ratio  increase  ?  Why  not  ? — Is  the  ratio 
of  the  halves  of  two  numbers  the  same  as  the  ratio  of  the  num- 
bers themselves  ?     Why  ? 

13-28.  Find  the  ratios  of  the  following  numbers  : 

Note. — Express  the  ratio  in  the  form  of  a  fraction,  and  reduce  to  lowest  terms. 


5: 

25  =  (' 

/25  =  75)     18 

54  = 

27 

108  = 

15:    25  = 

9  :  72  =s 

28 

42  = 

17 

93  = 

33  :  132  = 

3:33  = 

16 

96  = 

23 

69  = 

31  :  124  = 

6:  72  = 

19 

104  = 

14 

98  = 

13  :  117  = 

29-52. 

Find  the  ratios  of  the  following  fractions  : 

% 

Vs   = 

78:     %~ 

0.6    :0.12  = 

0.16:    0.4    = 

% 

V.  = 

%.:!%« 

0.5    :0.05  = 

0.33:    0.3    = 

% 

Vs    = 

»%:*%* 

0.35:0.07  = 

3.5    :    6.5    = 

V« 

%•« 

233/8:8V3  = 

27.2    :3.6    = 

2.6    :10.4    = 

% 

%  = 

4%  :  8%  = 

3.35:0.07  = 

8.45:10.25  = 

V* 

V.  = 

*y«t*% 

= 

0.26  :  0.( 

)7  = 

0.01  :    0.5    = 

PROPORTION. 

34£ 

Fill  the  blanks  in 

1.  42  :  —  =  7 

2.  —  :  19  =  3 

3.  18  :  %  =  - 

4.  27  :  -  =  % 

0 

the  following  statements' : 

5.  4:  —  =  y8              9.   10.4 

6.  %:-  =  %             10.  23% 

7.  3  :  15  =  —             11.       10 

8.  36:  —  =  7%          12.      — 

—  =4 

-  =  2% 
:—   =2 
:3     =9 

Suggestion. — To  fill  the  blank  in  the  first  problem  the  pupil  has  to  answer  the 
question,  "  42  is  1  times  what  number  ?  "  The  second,  "  What  number  is  3  times 
19?"  The  third,  "  18  is  how  many  times  lft  ?  "  The  fourth,  "  27  is  1/a  of  what 
number  ?  " 

From  the  foregoing  definitions  and  exercises  we  may  derive 
the  following 

Rules. 

353.  Mule,  —  1.  Multiply  the  consequent  by  the  ratio;  the 
product  will  be  the  antecedent. 

2.  Divide  the  antecedent  by  the  ratio ;  the  quotient  will  be 
the  consequent. 

13-18.  Prove  the  following  proportions  to  be  correct  : 
3:4::  6:    8  3  :    8     ::    6  :  16  % :    %  ::     % :    1% 

2:8::  6:24        1.05:    8.4::    1:    8  0.05:7      ::    0.3:42 

19-33.  Fill  the  blanks  in  the  following  statements,  determin  ■ 
ing  the  ratios  from  the  complet3d  couplets  : 


7 

56 

:  —  : 

16 

20 

5 

:  —  : 

2 

15 

— 

:    6: 

18 

— 

21 

::  56 

8 

21 
/3 

— 

:    6: 

9 

— :    3: 

:  16: 

5:  — 

:    8: 

3/4:    1 

:  —  : 

%:    1 

•:  60: 

17:51 

::  —  : 

9 

—  :    5: 

:3% 

6% 

64 

18:    6: 

:    21 

— 

12 

V,  :  %  : 

:    — 

24 

— 

0.2  :  —  : 

:      6 

108 

48 

10%:-: 

:    19 

95 

From  the  preceding  principles  and  exercises  we  derive  the 
following 

354-.   Rules  for  finding  the   Missing  Term  of  a  Proportion. 

Rule, — 1.  Find  the  ratio  of  the  complete  couplet ;  then, 

2.  If   the   antecedent    of  the   incomplete  couplet   be  wanting, 
multiply  the  consequent  by  the  ratio ;  or, 

3.  If  the  consequent  be  wanting,  divide  the  antecedent  by  the 
ratio. 

4.  The  result  will  be  the  term  required. 


350  STANDARD  ARITHMETIC. 

EXERCISES     IN     PROPORTION 

1.  If  24  hats  cost  $44,  what  will  150  hats  cost  ? 

Note. — To  avoid  error  in  the  statement  of  a  proportion,  an  arrangement  of  the 
terms  of  the  question  such  as  the  following  is  recommended : 

Complete  couplet.  Incomplete  couplet. 

?      is  the  cost  of  150  hats 
if  $44         "       "         24    " 

It  matters  not  whether  the  complete  or  the  incomplete  couplet  is  placed  first, 
nor  whether  the  sign  for  the  wanting  term  be  the  first  or  last  of  the  proportion ;  but 
it  is  essential,  in  all  questions  in  which  more  requires  more  and  less  requires  less, 
that,  if  the  upper  term  be  taken  as  the  antecedent  in  one  couplet,  the  same  should 
be  done  with  the  other. 

In  a  problem  in  which  more  requires  less,  and  less  requires  more,  it  is  neces- 
sary to  invert  the  terms  of  the  complete  couplet  in  the  statement.     For  example 

2.  If  it  requires  7  men  to  build  a  wall  in  27  days,  in  how  many 
days  would  9  men  perform  the  same  work  ? 

Preliminary  Arrangement. 
Days.  Men. 

?  j  9  Note. — An  arrow  pointing  downward  may 

27  v  7  be  used  to  indicate  terms  to  be  inverted. 

Since  9  men  require  less  time  to  do  the  work  than  1  men,  the  terms  of  the 
complete  couplet  are  inverted,  and  the  statement  is : 
Days.        Days.         Men.         Men.  w 

—    :    27    ::     7     :      9  27  X  |  =  21  days  Am. 

3.  How  many  tons  of  hay  will  325  acres  produce  if,  at  the 
same  rate,  13  acres  produce  40  tons  ? 

4.  What  time  would  it  require  for  7  men  to  mow  a  field,  if  3 
men  can  mow  it  in  3y3  days? 

5.  At  Christmas  8  eggs  were  sold  for  25^.  What  was  the 
cost  of  6  dozen  ? 

6.  Farmer  Black  pays  $52  %  rent  for  24  acres  of  land.  At 
the  same  rate,  what  will  he  have  to  pay  for  51  acres  ? 

7.  Mr.  H.  agrees  to  do  certain  work  in  15  days,  thereby  earn- 
ing $3.20  a  day.  How  much  will  he  earn  a  day  if  he  does  the 
work  in  10  days  ?    In  13  %  days  ? 


PROPORTION.  351 

8.  In  canning  5  lb.  of  raspberries  3  lb.  sugar  are  needed. 
How  many  pounds  sugar  for  38  lb.  of  berries  ? 

9.  If  with  the  money  I  have,  I  can  buy  84  lb.  of  coffee  at  25^ 
a  lb.,  how  many  lb.  could  I  buy  for  the  same  money  at  30^  a  lb.? 

10.  If  3  yd.  of  calico  cost  20^,  what  will  %  yd.  cost  ? 

Arrangement.  Statement.  Solution. 

?   is  the  cost  of  4/5  yd.  if         .      nn  *L  ,'.'"'' 

2(¥     I      I  £5yd.  ?:20::  «/.  :3  20  x  ^  =  « Vf  '-A* 

11.  If  wall  paper  be  20  inches  wide,  I  shall  need  7  rolls  to 
paper  a  room.  How  many  rolls  will  suffice  if  the  paper  be  24 
inches  wide  ?    If  30  inches  wide  ? 

12.  If  $750  will  yield  $120  interest  in  a  certain  time,  what 
interest  will  $600  yield  in  the  same  time  ? 

13.  A  man,  whose  step  measures  %  yard,  counts  1200  steps 
from  his  house  to  his  office.  How  many  steps  will  his  son  have 
to  take,  whose  step  measures  1/2  yd.  ? 


14.  If  each  man  on  board  ship  consumes  daily  \%/4  lb.  bread, 
their  bread  will  last  575  months.  How  much  will  each  man  get 
per  day  if  it  is  to  last  6  yg  months  ? 

15.  The  rate  of  two  pedestrians  is  as  5  :  4.  How  many  miles 
will  the  first  travel  in  the  same  time  in  which  the  second  travels 
84%  miles  ? 

16.  At  the  rate  of  $180  for  3/10  acre,  what  will  5  acres  cost  ? 

17.  The  heat  produced  by  a  cubic  yard  of  beech-wood  is  to 
that  produced  by  a  cu.  yd.  of  pine  as  9  :  7.  How  many  cu.  yd.  of 
beech-wood  are  needed  to  produce  the  heat  of  50  cu.  yd.  of  pine  ? 

18.  If  1%  yards  of  velvet  cost  15  Vj,  what  will  9  yd.  cost  ? 

19.  A  farmer  sowed  3  bu.  of  buckwheat  on  2%  acres.  How 
much  would  he  need  for  a  field  containing  4%  acres  ? 

20.  %  of  a  sum  of  money  is  $800.     How  much  is  %  of  it  ? 

21.  If  bread  is  7^  a  loaf  when  flour  is  sold  at  $6  a  barrel, 
what  should  flour  be  worth  when  bread  is  sold  at  8<fi  a  loaf  ? 


352  STANDARD  ARITHMETIC. 

Compound   Proportion. 

In  the  foregoing  exercises  the  ratio  for  the  incomplete  couplet 
is  found  from  one  complete  couplet  j  for  example  : 

1.  If  8  boys  can  pile  up  7  piles  of  cord  wood  in  a  day,  how 
many  boys  would  be  required  to  lay  up  21  piles  of  the  same  size 
in  the  same  time  ? 

Here  we  know  the  ratio  of  the  required  number  of  boys  to  the  given  number, 
from  the  fact  that  it  must  be  the  same  as  that  of  21  piles  to  7  piles.  Hence  the 
proportion  is : 

Statement. 

Boys.        Boys.         Piles.         Piles.  «   ■     ..  21  .  , 

p  g  21  7         Solution.— 8  X  --  =  24  boys. 

But  sometimes  the  ratio  of  the  incomplete  couplet  depends  on 
the  ratios  of  two  or  more  complete  ones  ;  for  instance,  let  prob- 
lem 1  be  changed  by  the  addition  of  the  words  printed  in  italics, 
as  follows  : 

2.  If  8  boys  can  pile  up  7  piles  of  cord  wood,  each  12  feet  long, 
in  1  day,  how  many  boys  could  pile  up  21  piles,  each  6  feet  long, 
in  the  same  time,  the  height  and  width  being  the  same  ? 

In  this  problem  the  number  of  boys  required  evidently  depends— first,  on  the 
number  of  piles  to  be  made ;  and,  second,  on  the  length  of  the  piles. 

The  way  to  indicate  this  dependence  of  a  term  on  two  or  more  ratios  is  to  write 
one  of  the  completed  couplets  under  the  other,  as  follows : 

?  :  8  ::  2J  j  jj        Solutions  x  f  X  I  =  12  boys. 

Taking  it  for  granted  that  wood  is  as  easily  piled  6  feet  high 
as  4  feet,  the  question  may  be  again  extended,  as  follows  : 

3.  If  8  boys  can  pile  up  7  piles  of  cord  wood,  each  pile  being 
12  feet  long  and  4  feet  high,  how  many  boys  can  pile  up  21  piles, 
each  6  feet  long  and  6  feet  high  9 

Here  the  number  of  boys  required  depends — first,  on  the  number  of  piles ;  sec- 
ond,  on  the  length  of  each  ;  and,  third,  on  the  height. 
This  dependence  is  indicated  thus : 


?:  8  :: 


21  : 

7 

6  : 

12 

91 

Solution.— 8  x  — 

6  : 

4 

7 

X^x|=18boys. 


PROPORTION.  353 

The  question  may  be  still  further  extended  by  introducing  the 
element  of  time  ;  thus  : 

4.  If  8  boys  can  pile  up  7  piles  of  cord  wood,  each  pile  12  feet 
long  and  4  feet  high,  in  2  dayst  how  many  boys  can  pile  up  21 
piles,  6  feet  long  and  6  feet  high,  in  3  days? 

In  this  example  the  number  of  boys  required  depends — first,  on  the  number  of 
piles ;  second,  on  the  length ;  third,  on  the  height ;  and,  fourth,  on  the  number  of 
days. 

When  there  are  so  many  conditions  in  a  problem,  it  is  convenient  to  make  a 
preliminary  arrangement  of  all  the  terms, ^is  follows: 

?  boys,    21  piles,     6  feet  long,  6  feet  high,  I  3  days. 
g     «         7     «      12     «      "     4    "       "     I 2      " 

It  is  to  be  observed  that  the  greater  the  number  of  days  the  less  will  be  the 
number  of  boys  required.  Hence  the  last  couplet  will  have  to  be  inverted  in  the 
statement. 

To  avoid  making  a  misstatement  in  such  cases,  the  couplet  which  is  to  be  in- 
verted should  be  distinguished  from  the  rest,  and  since  in  each  preceding  couplet 
the  upper  number  is  taken  for  the  antecedent,  an  arrow  may  be  placed  before  this 
one,  the  head  pointing  downward  to  indicate  that  here  the  lower  term  is  to  be  taken 
as  the  antecedent. 

The  statement  will  then  be : 

21  :     7 

6  :  12        Solution.— 8  X^X-|x^x|  =  12  boys. 
6:4  7       12      4      3  J 

2  :     3 

Note. — The  pupil  will  observe  that  the  preliminary  statement  is  all-sufficient 
for  the  solution,  care  being  taken  to  invert  the  terms  whose  ratios  are  inverse. 


Definitions. 

355.  A   Simple  Proportion  is   an   equality  of  two  simple 
ratios.     Thus,  8  :  32  ::  9  :  36  is  a  simple  proportion. 

356.  A   Compound  Proportion    is   an    equality  between   a 

simple  and  a  compound  ratio  or  between  two  compound  ratios. 

Thus, 

:«     0       21  :  7  ,6:42:6  ,  ,. 

**  :  **  ::    l  .         a™   o         ::  are  compound  proportions. 


354  STANDARD  ARITHMETIC. 

EXERCISES     IN     COMPOUND     PROPORTION. 

1.  Five  clerks  use  25  quires  of  paper  in  8  days.  At  the  same 
rate,  how  much  paper  will  6  clerks  use  in  10  days  ? 

2.  Six  lamps  consume  2  gallons  of  petroleum  in  8  days.  How 
many  lamps  will  consume  3  gal.  in  12  days  ? 

3.  Two  workmen  dig  a  ditch  of  24  yd.  in  length  and  3  ft.  in 
width  in  5  days.  How  long  will  it  take  3  workmen  to  dig  a  ditch 
30  yd.  long  «and  4  ft.  wide  ? 

4.  Eight  persons  spend  $296  on  a  journey  of  7  days.  How 
long  will  $300  last  7  persons  at  that  rate  ? 

5.  If  a  block  of  marble  5  ft.  long,  3  ft.  wide,  2  ft.  thick, 
weighs  4850  lb.,  what  will  a  block  weigh  measuring  7  ft.  in  length, 
4  ft.  in  width,  and  3  ft.  in  thickness  ? 

6.  Ten  cwt.  of  merchandise  cost  |3%  freight  for  245  miles. 
What  will  5  cwt.  cost  for  210  miles  ? 

7.  If  $700  at  interest  amounts  to  $770  in  15  months,  what 
sum  must  be  put  at  the  same  rate  to  amount  to  $845  in  the  same 
time  ? 

8.  From  the  milk  of  20  cows,  each  giving  18  qt.  daily,  16  '/j 
cheeses  of  50  lb.  each  are  made  in  42  days.  How  many  cows, 
giving  but  16  qt.  daily,  will  be  needed  to  make  33  cheeses  of  60 
lb.  each  in  28  days  ? 

9.  Being  asked  to  find  the  number  of  bricks  in  a  wall  10  ft. 
high,  922  ft.  long,  and  16  in.  thick,  I  found  that  a  part  of  the 
wall,  4  ft.  high,  4  ft.  long,  and  16  in.  thick,  contained  448 
bricks.     How  many  in  the  whole  wall  ? 

10.  Being  asked  to  find  the  probable  cost  of  a  lot  on  Seventh 
Street,  52  ft.  front  and  98  ft.  deep,  I  thought  of  my  own  lot  close 
by,  which  is  24  ft.  front  and  75  ft.  deep,  and  which  cost  me 
$1680.     What  should  the  answer  be  ? 

11.  If  450  copies  of  a  book  containing  300  pages  require  12 
reams  of  paper,  how  much  paper  will  be  needed  to  print  1500 
copies  of  a  book  of  170  pages  ? 


PROPORTION.  355 

Miscellaneous  Problems  in  Proportion; 

1.  Seven  men  need  16  days'  time  to  repair  a  dam.  How 
many  men  will  be  required  if  the  work  must  be  completed  in  14 
days  ? 

2.  The  prices  of  rye  and  wheat  are  to  each  other  as  5  :  6. 
What  is  the  price  of  wheat  if  rye  sells  at  80^  ?    At  75^  ? 

3.  For  7/10  yd.  lace  a  lady  pays  $5.60.  At  the  same  rate, 
what  does  the  merchant  ask  for  the  whole  piece  of  1(5  yd.  ? 

4.  The  spire  of  Nicolai  Church  at  Hamburg  throws  a  shadow 
of  49  yd.  in  length  when  a  vertical  staff  2yg  yards  high  throws 
a  shadow  of  245/308  yd.     What  is  the  height  of  the  spire  ? 

5.  John  takes  1200  paces  in  a  mile.  How  many  paces  must 
Harry  make,  who  makes  9  paces  to  every  8  of  John's  ?  What 
would  be  the  ratio  of  John's  paces  to  Harry's  ? 

6.  A  pavement  was  supposed  to  be  329  ft.  long,  but  the 
measuring-line  being  found  50  ft.  8%  in.  instead  of  50  ft.,  it  is 
required  to  find  the  true  length  of  the  pavement  without  another 
measurement. 

7.  A  bag  of  coffee  was  supposed  to  weigh  250  lb.,  but  the  50 
lb.  weight  used  in  weighing  was  really  50  lb.  5  oz.  Find  the  true 
weight  of  the  coffee. 

8.  The  liter  of  the  metric  system  =  1.0567  qt.  What  will 
represent  metrically  the  contents  of  a  gallon  ? 

9.  A  gram  is  equal  to  0.03527  oz.  Avoirdupois.  What  will 
represent  metrically  the  weight  of  a  pound  ? 

10.  If  a  meter  is  39.37  inches,  what  is  the  length  of  a  yard  by 
the  metric  system  ? 

11.  If  a  man  owes  $15,850,  and  has  but  $9750  to  pay  it  with, 
what  will  a  creditor  receive  to  whom  $1200  are  due  ? 

12.  If  a  wheel,  6y4  ft.  in  circumference,  turns  884.8  times  in 
going  a  given  distance,  how  many  times  will  a  wheel,  9y2  ft.  in 
circumference,  turn  in  going  the  same  distance  ? 


356  STANDARD  ARITHMETIC. 

13.  If  8y4  lb.  butter  will  pay  for  4%  lb.  tea,  how  much  butter 
will  pay  for  100  lb.  tea  ? 

14.  If  $360  gain  $40.80  in  1  yr.  5  mo.,  what  sum  will  $480 
gain  in  2  yr.  10  mo.  at  the  same  rate  ? 

15.  A.,  B.,  and  C.  are  partners.  A.  invests  $5050  in  the  busi- 
ness of  the  firm,  B.  $7070,  and  C.  $3030.  They  gain  $4545. 
What  is  the  share  of  each  in  the  gain  ? 

Suggestion. — Each  man's  share  of  the  gain  is  to  the  whole  gain  as  each  man's 
stock  is  to  the  whole  stock. 

16.  Three  men  freight  a  steamer  :  the  first  puts  $24,000  worth 
of  merchandise  aboard ;  the  second,  $18,000  worth ;  the  third, 
$15,000  worth.  During  a  storm  $1900  worth  of  the  freight  is 
thrown  overboard.     What  is  each  man's  share  of  the  loss  ? 

Suggestion. — Each  man's  share  of  loss  is  to  the  whole  loss  as  each  man's 
freight  is  to  the  whole  freight. 

17.  If  it  takes  21.78  paving  blocks  to  pave  5  rods  square, 
how  many  □.  rods  will  1,197,900  blocks  cover  ? 

18.  I  sent  to  my  agent  $2575  to  invest  after  deducting  his 
commission  of  3$.  What  was  his  commission  ?  What  sum  did 
he  invest  ? 

19.  I  send  to  my  broker  $1224  for  him  to  invest  after  deduct- 
ing his  commission  of  2$.  What  does  he  invest  ?  What  is  his 
commission  ? 

20.  When  a  bushel  of  wheat  was  sold  for  $1  the  price  of  a  loaf 
of  bread  was  5^.  What  will  be  the  price  of  a  loaf  of  equal  weight 
when  wheat  is  sold  at  4/6  dollar  a  bushel  ? 

21.  If  it  costs  $380  to  construct  a  wall  90  feet  long,  10  feet 
high,  and  16  inches  thick,  when  labor  is  worth  $2  per  day  of  10 
hours  each  ;  what  will  it  cost  to  build  a  wall  40  yards  long,  8  feet 
high,  and  12  inches  thick,  when  the  price  of  labor  is  $L80  per 
day  of  8  hours  each,  and  bricks  are  worth  $9  per  M  ?      Apply 

ratios  to  cost  of  construction,  i.  e.,  labor  ($3S0).     Cost  of  brick  found  separately 
(see  page  263). 


CHAPTER    XVIII. 

SQUARES   AND    CUBES. 

357.  A  square  measuring  3  in.  on  each  side  contains  9  sq.  in. ; 
hence  9  is  said  to  be  the  square  of  3.  For  a  like  reason,  16  is 
said  to  be  the  square  of  4,  25  of  5,  y4  of  1/2,  etc. 

Find  the  squares  of 

2  3  7  9  V2 


0.8 


% 


7a 


% 


.05 


It  may  seem  strange  to  the  pupil  that  the  square 
of  */2  *3  1Uy  hut,  to  convince  himself  that  it  is  so, 
he  has  only  to  draw  a  square  having  a  side  one  inch 
in  length,  and  to  divide  it  into  four  equal  parts,  as 
the  figure  in  the  margin.  He  will  readily  see  that  a 
square,  measuring  L/2  inch  on  each  side,  will  con- 
tain but  x/4  of  a  square  inch. 

In  the  same  way  illustrate  the  squares  of  .3,  1/5, 

/3>      U' 

358.  A  cubic  block,  measuring  3  in.  on  the  edge,  contains 
27  cubic  inches,  hence  27  is  said  to  be  the  cube  of  3.  For  a  like 
reason,  64  is  said  to  be  the  cube  of  4,  %  of  y2,  etc. 

What  are  the  cubes  of 

1.  .1  %  */.  5.  .3 


&  sq.  in. 

% 


7« 


ft 


% 


% 


Note. — The  cubes  of  integers  and  fractious  should  be  frequently  illustrated  by 
the  use  of  modeling  clay,  cubic  blocks,  etc. 

359.  To  square  a  number,  we  multiply  the  number  by  itself ; 

that  is,  we  use  it  twice  as  a  factor  to  produce  the  square.     Hence, 

the  square  of  a  number  is  also  called  its  second  power. 
10 


r 

358  STANDARD  ARITHMETIC. 

360.  To  cube  a  number,  we  multiply  the  square  of  the  num- 
ber by  the  number  itself  ;  that  is,  we  use  the  number  three  times 
as  a  factor  to  form  the  cube.  Hence,  it  is  said  that  the  cube  of 
a  number  is  its  third  power. 


Definitions. 

361.  A  power  of  a  number  is  the  number  itself,  or  the 
product  obtained  by  the  use  of  the  number  two  or  more  times 
as  a  factor.     The  number  itself  is  called  the  root  of  the  power. 

362.  The  number  of  times  a  root  is  employed  as  a  factor  is 
indicated  by  an  exponent,  which  is  commonly  a  small  figure 
written  to  the  right  and  a  little  higher  than  the  root. 

72  indicates  the  second  power  of  7,  hence  72  =  49. 

73  indicates  the  third  power  of  7,  hence  73  =  343. 

363.  The  process  of  raising  a  number  to  any  required  power 
is  called  Involution. 

The  process  of  involution  is  a  process  of  simple  multiplication  ; 
therefore  no  rule  is  necessary,  except  that  the  root  is  to  be  used 
as  a  factor  as  many  times  as  there  are  units  in  the  exponent. 


EXAMPLES      FOR      PRACTICE. 

Raise  the  following  numbers  to  the  powers  indicated: 

1.  17 3  4.  38 2  7.  12 3  10.  33 3  13.  2.152 

2.  3.22           5.  1.33           8.  0.13            11.  1282          14.  (5V3)2 
3-  CA)3            6.  (Ye)2           9.   (%)3            12.  325 2           15.  (7.iy4)3 


Find  the  values  of 

16.  122  X  2  18.  (72  X  33)  +  10  20.  (92  -4-  33)  X  72 

17.  92  X  22  19.  33  X  (42+  2)  21.  -(5Jx2) +  10 
Raise  to  the  second  power : 

22.  97  24.  35  26.  128  28.  826  30.  5287 

23.  98  25.  47  27.  371  29.  981  31.  6520 


SQUARES  AND  CUBES.  359 

Write  answers  to  the  following  questions  : 

1.  How  many  places  are  there  in  the  second  power -of  a  num- 
ber expressed  by  one  figure  ?  By  two  figures  ?  By  three  figures  ? 
By  four  figures  ? 

2.  Does  the  power  always  contain  twice  as  many  figures  as 
there  are  in  the  root  ?  Does  it  ever  contain  more  than  twice  as 
many  ?    May  it  contain  less  ? 

Suggestion. — To  answer  the  foregoing  questions  correctly,  and  with  confidence, 
the  pupil  should  square  the  greatest  and  the  smallest  numbers  that  may  be  ex- 
pressed by  one  figure  (1  and  9),  by  two  figures  (10  and  99),  etc. 

3.  How  many  decimal  places  are  there  in  the  second  power 
of  a  decimal  expressed  by  one  figure  ?  By  two  figures  ?  By  three 
figures  ?  By  four  figures  ?  In  the  square  of  a  decimal,  should  we 
ever  have  more  than  twice  as  many  decimal  places  as  there  are  in 
the  root  ?    Should  we  ever  have  less  ?    Why  not  ? 

Raise  to  the  third  power : 

1.  1  5.    10  9.    100  13.    1000  17.    10,000 

2.  9  6.    99  10.    999  14.    9999  18.    99,999 

3.  .1  7.  .01  11.  .001  15.  .0001  19.  .00001 

4.  .9  8.  .99  12.  .999  16.  .9999  20.  .99999 

Note. — The  square  of  999  may  be  conveniently  found  by  subtracting  999  from 
999,000.  Why  ?  The  same  method  may  be  applied  to  any  other  number  expressed 
by  9's. 

Write  answers  to  the  following  questions  : 

1.  How  many  places  in  the  third  power  of  an  integer  expressed 
by  one  figure  ?  By  two  figures  ?  By  three  figures  ?  By  four 
figures  ? 

2.  How  many  times  as  many  figures  in  the  power  as  there  are 
in  the  root  ?  Are  there  ever  more  than  three  times  as  many  ? 
Are  there  ever  less  ?    How  many  less  may  there  be  ? 

3.  How  many  places  are  there  in  the  third  power  of  a  decimal 
of  one  place  ?  Of  two  places  ?  Of  three  places  ?  etc.  In  the  cube 
of  a  decimal,  should  we  ever  obtain  either  more  or  less  than  three 
times  as  many  decimal  places  as  there  are  in  the  root  ?    Why  not  ? 


360  STANDARD  ARITHMETIC. 

364-.  Another  method   of  raising  numbers  to  the  second  and 

third  power. 

It  will  be  found  useful  to  note  carefully  how  the  tens  and  units  are  combined 
in  the  process  of  raising  a  number,  expressed  by  two  or  more  figures,  to  its  second 
or  third  power.     For  illustration,  let  us  take  the  example : 

Kaise  43  to  its  second  power. 

Analytical  Method. 

40  +3 

40 +3 

40  X  3  +        3X3 

40  X  40       +  40  X  3 

40  X  40       +       2  X  (40  X  3)       +       (3  X  3) 

sq.  of  the  tens.  twice  the  tens  by  the  units.  sq.  of  the  units. 

Little  explanation  is  here  necessary,  except  that,  in  the  analytical  method, 
instead  of  actually  multiplying  and  adding,  as  in  the  common  process,  we  only  indi- 
cate the  multiplications  and  additions  by  means  of  signs.  This  we  do,  that  we  may 
trace  the  tens  and  units  separately  through  the  process,  to  see  where  we  may  find 
them  in  the  product.     Thus,  in  this  case  we  sec  that 

365.  The  square  of  Jf.3  is  equal  to  the  square  of  the  tens,  plus 
twice  the  product  of  the  tens  by  the  units,  plus  the  square  of  the 
units. 

It  may  be  shown  that  the  same  is  true  of  any  number. 

Raise  the  following  numbers  to  the  second  power,  using  the  ana- 
lytical method : 

21.  26  25.  64  29.  76 

22.  35  26.  78  30.  23 

23.  45  27.  59  31.  54 

24.  53  28.  82  32.  83 


33. 

94 

34. 

72 

35. 

63 

36. 

49 

40  s 

1600 

2 

(40 

xq; 

= 

720 

9* 

81 

Note. — The  pupil  should  be  able  to  perform  these 
operations  without  the  aid  of  the  pencil.  For  the  last 
he  would  say  1600,  720,  2320,  2401  the  square  of 
49.  If  required  to  write  out  the  work,  he  would  write 
as  in  the  margin.  49  2401 

A  like  process  may  be  used  in  raising  numbers,  expressed  by 
three  or  more  figures,  to  the  second  power. 


SQUARES  AND  CUBES.  361 

Example. — l.  Raise  493  to  the  second  power. 

Tlie  square  of  490  is  obtained  by  annexing  two  ciphers  to 

the  square  of  49,  as  already  found =  240100 

Twice  the  product  of  490  x  3 =      2940 

The  square  of  the  units =  9 

The  square  of  493 243049 

Note. — The  pupil  will  do  well  to  familiarize  himself  with  this  process,  not  for 
its  own  sake,  but  that  he  may  be  the  better  prepared  for  the  demonstration  of  the 
process  of  extracting  the  square  root,  which  is  exactly  the  reverse  of  this. 

In  like  manner  raise  the  following  numbers  to  the  second  power : 
2.  528  3.  732  4.  236  5.  429  6.  523 

366.  If  a  number  be  divided  into  any  two  parts,  it  may  be 
shown  that  the  square  of  the  whole  number  is  equal  to  the  square 
of  the  first  part  -f-  twice  the  product  of  the  first  by  the  second  + 
the  square  of  the  second. 

Example. — 7.  Find  the  square  of  16. 

Solution. — 16  =  7  +  9  ;  according  to  the  formula,  therefore, 
16  2  =49  +  126  +  81  =256. 

In  the  same  way  compute  the  squares  of  the  following  numbers  : 

8.  17  10.  35  12.  81  14.  126  16.  839 

9.  23  11.  46  13.  94  15.  348  17.  476 
"We  shall  find  it  useful  to  observe  also  how  the  tens  and  units 

of  the  root  are  combined  in  its  third  power. 

Process  of  computing  the  Third  Power  analyzed. 

Example.— What  is  the  cube  of  43  ? 

Solution. — Multiplying  the  square  of  43,  as  already  found,  by  40  +  3,  we  have 
(40X40)        +2(40X3)        +(3x3) 

40         +      3 

(40X40X3)  +  2  (40X3X3)  +  (3x3x3) 

(40X40X40)  +  2  (40X40X3)  +      (40x3x3) 

(40X40X40)  +  3  (40X40X3)  +  3  (40x3x3)  +  (3X3X3)  \ 

cu.  of  tens.  3  x  sq.  of  tens  by  units.     3  x  sq.  of  units  by  tens.     cu.  of  units,    r  — 

Or,  64000        +  14400.       +  1080        +         27        ) 


362  STANDARD  ARITHMETIC. 

Whereby  we  find  that 

367.  The  cube  of  43  is  equal  to  the  cube  of  the  tens  +  three 
times  the  square  of  the  tens  multiplied  by  the  units  +  three  times 
the  square  of  the  units  multiplied  by  the  tens  +  the  cube  of  the 
units. 

The  same  may  be  shown  to  be  true  of  any  number  whatsoever. 

In  like  manner  find  the  cubes  of 

1.  36  2.  27  3.  92  4.  85  5.  73  6.  95 

Note. — Test  the  accuracy  of  these  results  by  the  common  process. 

7.  Calculate  the  cube  of  47,  and  write  the  solution  in  the  fol- 
lowing form  : 

The  cube  of  the  tens 40x40x40=    64000 

Three  times  the  square  of  the  tens  by  the  units.   3  x  40  x  40  x     7  =    33600 
Three  times  the  tens  by  the  square  of  the  units.   3x40x     7  x     7=      5880 

The  cube  of  the  units 7x     7x     7  =        343 

103823 

Write  out  the  solution  of  the  following  examples  in  the  same 
way: 

Find  the  third  powers  or  cubes  of 

8.  37  10.  45  12.  65  14.  83  16.  28 

9.  54  11.  23  13.  71  15.  92  17.  74 

368.  If  a  number  be  divided  into  any  two  parts  whatsoever, 
it  may  be  shown  that  the  cube  of  the  first  part  +  three  times  the 
square  of  the  first  multiplied  by  the  second  +  three  times  the 
square  of  the  second  by  the  first  +  the  cube  of  the  second  is 
equal  to  the  cube  of  the  number  itself. 

Example. — Find  the  cube  of  82. 

Solution. — 82  =  28  +  54 ;  according  to  the  formula,  therefore,  we  have 
82  »  =  (28  +  54)  3  =  28  3  +  3  (28  2  x  54)  +  3  (28  x  54  2)  +  54  3,  or, 

Cube  of  the  first 28  3  =    21952 

Three  times  the  square  of  the  first  by  the  second  . .  3  x  28  2  x  54     =  127008 

Three  times  the  square  of  the  second  by  the  first  . .  3  x  28     x  542  =  244944 

Cube  of  the  second 543  =  157464 

Third  power  or  cube  of  82 ==  551368 


SQUARES  AND  CUBES  363 

Definitions. 

369.  Evolution  is  a  process  of  finding  the  root  of  a  given 
number. 

Evolution  is  the  converse  of  Involution.  In  the  latter  the  root  is  given  to 
find  the  power ;  in  the  former  the  power  is  given  to  find  the  root. 

370.  Square  root  is  indicated  by  the  sign  */ y  thus,  Vl6  =  4 
is  read,  "The  square  root  of  16  is  equal  to  4."  The  cube  root 
is  indicated  by  the  same  sign  with  the  aid  of  a  small  figure  3 
placed  above  it ;  thus,  V'27  =  3. 

371.  The  sign  of  evolution  (yO  is  called  the  radical  sign, 
from  the  word  radix,  which  means  root.  The  figure  which  in- 
dicates the  degree  of  the  root  to  be  extracted  is  called  the  index 
of  the  root. 

372.  The  root  of  a  number  is  indicated  also  by  a  fractional 
exponent,  the  denominator  of  which  is  the  index  of  the  root. 
Thus,  16*  means  the  same  as  a/16.  27^  is  only  another  ex- 
pression for  a/27. 

To  find  the  Square  Root  of  a  Number. 

Example. — l.  Let  it  be  required  to  extract  the  square  root 
of  1849. 

It  is  to  be  remembered  that  the  square  of  any  number  ex- 
pressed by  tens  and  units  is  equal  to 

The  square  of  the  tens  -\-  twice  the  product  of  the  tens  by  the 
units  +  the  square  of  the  units. 

Let  it  also  be  remembered  that 

No  part  of  the  square  of  tens  can  be  found  in  tens  or  units' 
place,  and  that  no  part  of  the  product  of  tens  and  units  can  be 
found  in  units9  place. 

The  process  of  extracting  the  root  consists  in  obtaining  the  tens  of  the  root 
from  the  square  of  the  tens,  and  the  units  of  the  root  from  the  remaining  parts  oi 
the  power. 


364  STANDARD  ARITHMETIC. 

First  Step. — To  find  the  tens9  figure  of  the  root.—  Since  the 
square  of  the  tens  can  not  contain  anything  less  than  hundreds, 
the  two  figures  at  the  right  con- 
tain no  part  of  the  square  of  the 

tens,  and  are  therefore  disregard-  1849  |  40  +  3  =  43 

ed  for  the  present.     The  greatest         40  x  40  =  1600  ~~ 
square  in  1800  is  1600,  the  square  249 

root  of  which  is  40.  80  x    3  =    240 

We  place  40  to  the  right  of  3x3=       9 

the  given   number   and  subtract 
1600  (40  X  40)  from  1849. 

Second  Step. — To  find  the  units'  figure  of  the  root. — Since  no 
part  of  the  product  of  tens  by  any  integer  whatsoever  can  be  less 
than  ten,  the  right-hand  figure  of  the  remainder  can  contain  no 
part  of  the  product  of  the  tens  by  the  units,  and  hence  it  is  dis- 
regarded for  the  present.  The  24  tens  then  being  twice  the  prod- 
uct of  the  tens  by  the  units,  we  obtain  the  units  by  dividing  24 
by  twice  the  tens.  The  quotient  is  3,  which  we  add  to  the  40. 
Multiplying  twice  the  tens,  or  80,  by  3,  we  have  twice  the  prod- 
uct of  the  tens  by  the  units,  and  subtracting  this  we  have  re- 
maining only  the  square  of  the  units.  Subtracting  the  square  of 
the  units  nothing  remains,  and  thus  43  is  found  to  be  the  square 
root  of  1849. 

Notes. — 1.  Instead  of  multiplying  80  and  3  separately  by  3  1849  |  43 

and  adding  the  products,  the  work  is  somewhat  shortened  by  mul-  ]  qqq  ~~ 

tiplying  the  sum  of  80  and  3  by  3.     The  work  would  then  stand  831i249 

as  in  the  margin. 

249 

2.  When  the  number  whose  root  is  to  be  found  is  expressed  

by  five  or  six  figures,  and  it  is  thus  known  that  the  root  must 
contain  three  figures,  the  work  may  be  commenced  with  the  two  left-hand  periods 
as  if  they  were  the  only  ones,  and  when  the  root  of  these  has  been  so  obtained  the 
operation  may  be  completed  as  though  the  two  figures  of  the  root  already  found 
were  so  many  tens,  as  they  really  are. 

3.  The  only  difficulty  in  the  extraction  of  the  square  root  is  met  with  when,  on 
multiplying  twice  the  tens  +  the  units  by  the  units,  the  product  is  found  to  be  too 
great.  This  difficulty  arises  from  the  trial  divisor  being  sometimes  considerably 
increased  by  the  tens  that  come  from  the  square  of  the  units. 


SQUARES  AND  CUBES. 


365 


Thus,  in  the  example  here  given,  if  the  54  tens  expressed  by 
the  first  two  figures  of  the  remainder  were  the  product  of  only 
the  tens  by  the  units,  we  should  obtain  the  exact  number  of  the 
units  by  dividing  it  by  twice  the  tens.  But,  in  this  case,  we  should 
have  9  for  the  quotient,  which  is  evidently  too  great,  since  by 
forming  the  sum  of  the  product  of  twice  the  tens  +  the  units  by 
the  units  we  get  621,  which  is  greater  than  the  dividend. 

In  such  cases  as  this  we  have  to  diminish  the  units  of  the  root,  not  forgetting 
to  change  the  right-hand  figure  of  the  partial  divisor  at  the  same  time,  until  we 
obtain  a  product  not  greater  than  the  tens.  The  following  example  presents  an 
extreme  case  of  this  nature : 


1444  1  38 
9 
68)544 
544 


2.  Required  the  square  root  of  321735969. 


The  number  being  divided  into  periods  of 
two  figures  each,  the  first  is  3  (hundred  mill- 
ions). The  greatest  square  in  3  is  1.  Sub- 
tracting this,  and  annexing  the  next  period  to 
the  remainder,  we  have  22(1)  for  a  partial 
dividend,  and,  doubling  the  root  already  found, 
we  have  2  for  the  partial  divisor. 

But  the  next  term  of  the  root  can  not  be 
greater  than  9.  We  try  it,  and  obtain  a  re- 
sult too  great.  We  next  try  8,  and  again  the 
product  is  greater  than  the  partial  dividend. 
Finally,  by  trying  7,  we  obtain  a  product  less 
than  the  partial  dividend.  Having  subtracted 
this,  we  proceed  with  the  solution. 

Note. — In  this  example  we  have  also  a 
case  which  is  of  common  occurrence,  that  is, 
the  increase  of  the  tens  of  the  previous  divisor 
by  the  doubling  of  the  units.  The  pupil  can 
avoid  any  confusion  by  observing  that  the  en- 
tire part  of  the  root  already  found  is  always 
doubled  before  the  new  figure  of  the  root  can 
be  found. 


First  Trial. 
32i735969  |  19 


1 

29)221 
261 

Second  Trial. 

321735969  [  18 
1 

28)221 
224 

Third  Trial. 

321735969  [  17937 

1 
27)221 

189 
349)3273 
3141 
3583)13259 


A  like  difficulty  will  be  found  in  each  one  of  the  following  : 

Find  the  square  roots  of 

3.  310993225  4.  738534976  5.  27950824225 

In  the  foregoing  examples  some  of  the  remainders  are  large. 
A  few  are  here  given  in  which  some  very  small  remainders  occur. 


3(36  STANDARD  ARITHMETIC. 

6.  Let  it  be  required  to  extract  the  square  root  of  731864809. 

731864809  |  27053 

On   bringing  down  the  third  period,  we  4 . 

find  that  the  dividend  does  not  contain  twice  47)331 

the  root  already  found ;   in  such  a  case  we  329 
write  a  cipher  in  the  root  and  also  one  to  the 
right  of    the  divisor.     We  then   bring  down 
the  next  period  and  proceed  as  before. 


5405)28648 
27025 
54103)162309 
162309 


Find  the  first  powers  or  roots  of  the  following  squares  : 
7.  1855197184  8.  36125464489  9.  4901120064 

Rule  for  extracting  the  Square  Root.' 

373.  Mule. — 1.  Separate  the  given  number  into  periods  of  two 
figures  each  by  placing  a  dot  over  the  units'  place  and  every 
second  figure  to  the  left,  and  in  case  of  decimals  to  the  right  also, 
annexing  a  cipher,  if  necessary  to  complete  a  decimal  period. 

2.  Find  by  trial  the  greatest  square  in  the  left-hand  period, 
and  place  its  root  in  the  form  of  a  quotient  at  the  right. 

3.  Subtract  the  square  of  the  root  thus  found  from  the  first 
period,  and  to  the  remainder  bring  down  the  second  period  for  a 
dividend. 

4.  Double  the  root  already  found  for  a  trial  divisor ;  divide  the 
dividend  by  it,  and  write  the  quotient  for  the  second  term  of  the 
root. 

5.  Annex  the  second  figure  of  the  root  to  the  trial  divisor.  The 
result  will  be  the  complete  divisor.  Multiply  this  by  the  second 
term  of  the  root,  and  subtract  the  product  from  the  dividend. 

6.  Repeat  the  operation  as  in  4  and  5  until  the  periods  are  all 
brought  down. 

Notes. — 1.  When  a  partial  divisor  is  not  contained  in  a  dividend,  annex  a  cipher 
to  the  root  already  obtained,  and  also  to  the  partial  divisor.  Bring  down  the  next 
period,  and  proceed  as  in  4  and  5. 

2.  When  the  given  number  is  not  a  perfect  square,  and  hence  a  remainder 
occurs  after  the  last  period  has  been  used,  one  or  more  periods  of  decimal  ciphers 
may  be  annexed,  and  the  operation  continued  as  before.  The  figures  in.  the  root 
corresponding  to  the  decimal  periods  will  be  decimals. 

3.  It  must  be  kept  in  mind  that  no  period  should  contain  an  integer  and  deci- 
mal, and  that,  if  there  is  an  odd  number  of  decimal  places  in  the  given  number,  the 
last  period  must  be  completed  by  annexing  a  cipher. 


SQUARES  AND  CUBES, 


367 


To  find  the  Square  Root  of  a  Common  Fraction. 

374-.  Mule, — 1.  Reduce  the  common  fraction  to  a  decimal,  and 
extract  the  square  root.     Or, 

2.  Extract  the  square  root  of  the  numerator  and  of  the  de- 
nominator.   The  result  will  be  the  terms  of  the  root. 

Note. — If  only  the  denominator  of  the  fraction  is  a  perfect  square,  the  latter 
is  the  more  convenient  method.  If  the  denominator  is  not  a  perfect  square,  it  may 
be  made  so  by  multiplying  both  terms  of  the  fraction  by  the  denominator. 


EXAM  PLES. 

Find  the 

square  root  of 

1.  36864 

5.  244036 

9.  579121 

13.  966289 

2.  81225 

6.  258064 

10.  734449 

14.  1081600 

3.  168921 

7.  396900 

11.  820836 

15.  1177225 

4.  212521 

8.  499849 

12.  950625 

16.  1234321 

Find  one 

of  the  two  equal  factors  of 

17.  6838225 

20.  296356225 

23.  44502241 

18.  9048064 

21.  3196944 

24.  61685316 

19.  6885376 

22.  19228225 

25.  179586801 

Extract  the  square  root  of 

26.  .0961 

30.  28867 

34. 

3819.24 

38.  5416.96 

27.  15.21 

31.  33489 

35. 

1.338649 

39.  50.1264 

28.  22.09 

32.  4.2849 

36. 

226.8036 

40.  .00720801 

29.  .0004 

33.  17.3056 

37. 

.00001024 

41.  290.225296 

Extract  the  square  root  of 

42.  5 

46.  2 

50.  20  % 

54.  3% 

43.  .5 

47.  .6 

51.  153% 

55.  35% 

44.  .05 

48.  26 

52.  1%9 

56.  27% 

45.  .005 

49.  .02 

53.  23.1 

57.  36% 

Find  the 

square  root  of 

58.  % 

fil      625/ 
•*•         /6T6 

64.  % 

67.  % 

59.  % 

fi9     3136/ 
*>^'           /5329 

65.  17% 

68.  5% 

60.  % 

fio     84681/    . 

66.  11% 

69.  38% 

368  STANDARD  ARITHMETIC. 

375.  To  find  the  Cube  Root  of  a  Number. 

Example. — l.  Let  it  be  required  to  find  the  root  of  which  42875 
is  the  third  power. 

It  must  be  kept  in  mind  that  the  cube  of  any  number  com- 
posed of  tens  and  units  is  equal  to 

The  cube,  of  the  tens  -f-  three  times  the  product  of  the  square  of 
the  tens  by  the  units  +  three  times  the  product  of  the  tens  by  the 
square  of  the  units  ~\-  the  cube  of  the  units. 

Since  the  cube  of  tens  can  never  fall  short  of  a  thousand,  we  shall  not  find  any 
part  of  it  in  the  first  three  figures  to  the  right,  hence  they  are  disregarded  in  finding 
the  tens. 

Evidently  the  root  of  42000  can  net  be  so  42875  |  3 

great  as  40,  since  the  cube  of  40  is  64000 ;  30  X  30  X  30  =  27000  ~ 

but,  the  cube  of  30  being  only  27000,  the  T^ftT^ 

real  root  must  be  between  30  and  40,  and 

hence  3  must  be  the  tens'  figure  sought  for.     Subtracting  the  cube  of   30  from 
42875  we  have  15875  remaining. 

But  having  taken  the  cube  of  the  tens  from  42728,  the  remainder  must 
contain 

(1.)  3  times  the  square  of  the  tens  x  the  units  + 
(2.)  3  times  the  tens  x  the  square  of  the  units  + 
(3.)  the  cube  of  the  units. 

Now,  since  we  know  that  the  tens'  figure  is  3,  and  hence  that  three  times  the 
square  of  the  tens  (3  times  30  x  30)  is  2700 ;  and  since  we  know  also  that  the 
remainder,  15875,  contains  the  product  of  this  2700  by  the  units,  we  next  try  to 
find  the  units  by  dividing  15875  by  2700.  But,  since  15875  contains  something 
more  than  3  times  the  square  of  the  tens  by  the  units,  our  quotient  is  very  likely 
to  be  too  great.  If  it  contained  nothing  more,  it  would  be  easy  to  find  the  units  by 
division.  Yet  we  may  be  sure  that  it  can  not  be  9,  nor  8,  nor  7,  nor  6,  since  6 
times  2700  alone  is  greater  than  15875.  The  figure  may  be  5,  but  we  can  not  be 
sure  that  even  5  is  not  too  great,  until  we  find  that  the  sum  of  the  three  items  is 
not  greater  than  15875.  Let  us  try  it,  however,  by  completing  the  work  as  if  it 
were  the  right  figure,  thus : 

3  times  the  square  of  the  tens  x  the  units. . .   3  x  30  x  30  x  5  =  13500 
3  times  the  tens  x  the  square  of  the  units. . .  3  x  30  x     5x5=    2250 

The  cube  of  the  units 5x     5x5=      125 

15875 

The  sum  of  all  these  parts  being  equal  to  the  remainder,  15875,  it  is  clear  that 
5  is  the  correct  units'  figure,  and  that  35  is  the  cube  root  of  42875. 


SQUARES  AND  CUBES. 


369 


The  whole  work  of  solu- 
tion may  be  put  into  this 
form : 


But,  instead  of  multiply, 
ing  three  times  separately  by 
the  factor  5,  and  adding  the 
products,  we  may  multiply  the 
sum  of  the  products  of  the  other 
factors  by  5  in  one  operation. 
The  work  would  then  stand  as 
here  jriven. 


30  X  30  x  30  = 

42875 
27000 

35 

3x30x30x5  =  13500 

3x30x  5x5=  2250 

5x5x5=   125 

15875 
15875 

30  x  30  x  30  = 

42875 
27000  ' 

35 

3  x  30  x  30  =  2700 

3  x  30  x  5  =  450 

5x5=   25 

15875 

3175  x  5 


15875 


Note. — If,  on  completing  the  work,  we  had  found  that  5  was  too  great,  we 
should  have  had  to  take  4  as  the  units'  figure,  and  possibly  even  that  might  have 
been  found  too  great.  In  that  case,  we  should  have  had  to  take  3,  and  try  again. 
The  finding  of  the  square  or  cube  root  of  a  number  is  often  a  process  of  guessing, 
and  testing  the  correctness  of  the  guess. 


2.  Find  the  cube  root  of  22906304. 


Here  the  partial  di- 
visor is  contained  11 
times  in  the  first  re- 
mainder, but  inasmuch 
as  we  know  that  the 
next  figure  of  the  root 
can  not  be  greater  than 
9,  we  try  9,  and,  finding 
it  to  be  too  great,  we  try 
8,  which  proves  to  be 
the  right  figure. 


28  = 

3  x  20  2  =  1200 

3  x  20  x  8  =480 

82=   64 


22906304  |  284 
8 


1744 


14906 


13952 


3  x  280  s  =  235200 

3  x  280  x  4  =   3360 

42  =    16 


238576 


954304 


954304 


3.  Extract  the  cube  root  of  28372625. 


On  bringing  down  the  second  period,  it  is  found  that  the  partial  divisor  (2700) 
is  not  contained  in  the 

28372625  |  305 
33=  27 


dividend,  hence  we  place 
a  cipher  in  the  root,  bring 
down  the  next  period, 
and  proceed  again  ac- 
cording to  the  rule  (4 
and  5). 


3  X  300  2  =  270000 

3  x  300  x      5     =      4500 

5x5=  25 

274525 


1372625 


1372625 


370  STANDARD  ARITHMETIC. 

Rule  for  extracting  the  Cube  Root. 

376.  Mule,  —  1.  Separate  the  given  number  into  periods  of 
three  figures  each  by  placing  a  dot  over  the  units'  place  and  every 
third  figure  to  the  left,  and,  if  there  be  decimals,  to  the  right  also. 
Annex  one  or  two  ciphers  if  necessary  to  complete  a  decimal 
period. 

2.  Find  the  greatest  cube  in  the  left-hand  period,  and  place  its 
root  at  the  right  for  the  first  term  of  the  root  sought. 

3.  Subtract  the  cube  of  the  first  term  of  the  root  from  the  first 
period,  and  to  the  remainder  annex  the  second  period  for  a  divi- 
dend. 

4.  Take  three  times  the  square  of  the  root,  already  found,  as 
a  trial  divisor,  ascertain  how  many  times  it  is  contained  in  the 
dividend,  and  annex  the  result  to  the  root  as  a  trial  term. 

5.  Find  the  sum  of 

3  times  the  square  of  the  first  term  of  the  root, 
+  3  times  the  product  of  the  first  by  the  trial  term, 
+  the  square  of  the  trial  term,  for 
a  complete  divisor.     Multiply  the    sum   by  the  trial  term  of  the 
root,  and  subtract  the  product,  if  not  too  great,  from  the  dividend. 

6.  If  the  product  be  greater  than  the  dividend,  diminish  the 
trial  term  of  the  root,  and  proceed  as  before. 

7.  The  remainder,  if  there  be  any,  with  the  next  period,  will 
form  another  partial  dividend  with  which  we  proceed  again,  as 
directed  in  4  and  5. 

Notes. — 1.  When  a  partial  divisor  is  not  contained  in  a  dividend,  annex  a  cipher 
to  the  root  obtained,  annex  two  ciphers  to  the  partial  divisor,  bring  down  the  next 
period,  and  proceed  as  directed  in  4  and  5. 

2.  When  the  given  number  is  not  a  perfect  cube,  and  hence  a  remainder  occurs 
after  the  last  period  has  been  used,  one  or  more  periods  of  decimal  ciphers  may 
be  annexed,  and  the  operation  continued  as  before.  The  figures  in  the  root  cor- 
responding to  the  decimal  periods  will  be  decimals. 


To  extract  the  Cube  Root  of  a  Common  Fraction. 

377.  Hule.—l.  Reduce  the  common  fraction  to  a  decimal,  and 
extract  the  cube  root. 

2.  Or,  if  the  numerator  and  denominator  are  perfect  cubes, 
extract  the  cube  roots  of  the  terms  separately.  The  results  will 
be  the  terms  of  the  root. 

Note. — If  only  the  denominator  of  the  given  fraction  is  a  perfect  cube,  the 
latter  is  the  more  convenient  method. 


SQUARES  AND  CUBES.  371 

-SLATE     EXERCISES. 

Find  the  cube  root  of 

1.  6859  4.  2406104  7.  49027896 

2.  12167  5.  3869893  8.  66430125 

3.  27000  6.  5545233  9.  929714176 
Extract  the  cube  root  of 

10.  1412467848  12.  3341362375  14.  3616805375 

11.  1865409391  13.  2857243059  15.  4065356736 
Find  the  cube  root  of 

16.  830.584  18.  1.092727  20.  .000175616 

17.  .970299  19.  .002197  21.  .007645373 

Find  the  cube  root  of  the  following  numbers,  carrying  incomplete 
roots  to  three  or  fve  decimal  places,  as  may  be  required : 

22.  1.  24.  .01  26.   .001  28.  %  30.  % 

23.  2.  25.  .02  27.  .002  29.  3/4  31.  % 


The  Extraction  of  Roots  of  Perfect  Powers. 

378.  The  Square  Root. — The  square  root  of  a  number  being  one  of  two 
equal  factors,  the  square  root  of  a  perfect  power  may  be  found  by  resolving  the 
power  into  its  prime  factors,  separating  them  into  two  identical  sets,  and  finding  the 
product  of  one  set. 

Example. — l.  Let  it  be  required  to  find  the  square  root  of  3136. 

The  prime  factors  of  3136  are  2,  2,  2,  2,  2,  2,  7  and  7.  These  are  separable 
into  two  sets  of  factors,  each  containing  2,  2,  2  and  7.  The  product  of  2  x  2  x  2  x 
7  =  56.     Hence,  56  is  the  square  root  of  3136.     In  like  manner  find  the 


2.  V484  3.  a/1024  4.  a/16384  5.  V234256 

379.  The  Cube  Root. — On  the  same  principle,  the  cube  root  of  a  perfect 
power  may  be  obtained  by  resolving  the  power  into  its  prime  factors,  separating 
them  into  three  identical  sets,  and  finding  the  product  of  one  set. 

Example. — 6.  Find  the  cube  root  of  13824. 

The  prime  factors  of  13824  are  2,  2,  2,  2,  2,  2,  2,  2,  2,  3,  3,  3.  These  are 
separable  into  three  sets  of  factors,  each  containing  3,  2,  2  and  2.  The  product  of 
3  x  2  x  2  x  2  is  24.     Hence  24  is  the  cube  root  of  13824.     In  like  manner  find  the 

7.  V3375    8.  V9261     9.  V35937    10.  V250047 


372  STANDARD  ARITHMETIC. 

Constructive   or  Geometric   Solution   of   the    Problem 
of  the   Square    Root. 

Problem. — Let  it  be  required  to  find  the  side  of  a  square  which 
shall  contain  3249  sq.  in. 

Solution. — Suppose  that  we  have  3249  pieces  of  card-board,  each  one  inch 
square,  and  let  it  be  required  to  arrange  them  so  that  they  shall  together  make  one 
complete  square.  The  number  of  pieces  on  one  side  will  be  the  length  of  the  side 
in  inches. 

First  let  us  try  one  hundred  pieces  to  the  side.  To  make  a  square,  we  must 
have  as  many  rows  of  square  pieces  as  there  are  pieces  in  a  row ;  hence,  with  this 
beginning,  we  would  need  10,000  pieces.  But,  inasmuch  as  we  have  not  so  many, 
we  plan  to  make  our  square  a  smaller  one.  Let  us  try  60.  This  we  would  find 
would  require  3600  pieces,  which  again  is  more  than  the  given  number.  Trying 
50,  we  find  that  we  can  complete  a  square  of  this  size,  and  have  some  pieces  left. 
The  required  square  will  therefore  contain  between  50  and  60  pieces,  and  hence  5 
must  be  the  tens'  figure  of  the  root. 

Having  completed  the  square  of  50  pieces  to  the  No  pjece(, 

side,  we  ascertain,  as  in  the  margin,  that  there  are  749  3249  I  5  • 

pieces  left  with  which  the  size  of  the  square  is  to  be  in-       Kn     KA  _  okaa  

creased. 

Laying  down  a  row  on  each  one  of  two  adjacent  *4y 

sides  (50  to  the  side) — for  we  must  increase  the  length 

and  breadth  equally — we  require  twice  50,  or  100  pieces.  If  we  add  two  rows  to 
each  side,  we  shall  require  2  times  100  pieces,  and.  if  three  rows  be  added,  we 
shall  require  3  times  100  pieces,  that  is,  without  filling  the  corner.  To  fill  the 
corner,  we  shall  need  as  many  more  pieces  in  each  row  as  there  are  additional 
rows.  Thus,  if  there  are  three  rows  added  to  each  side,  we  must  extend  each 
row  of  one  side  by  the  addition  of  3  pieces.  Thus  we  may  proceed,  making  suc- 
cessive additions  of  one  row  at  a  time  to  each  of  two  sides  until  all  the  pieces  are 
taken,  or  until  the  largest  possible  square  is  made  out  of  the  3249  pieces  given. 
But  the  process  can  be  somewhat  shortened  by  ascertaining  at  once  how  many  rows 
are  to  be  added.  This  we  can  do  very  nearly  by  dividing  749  by  the  number  of 
pieces  in  two  rows,  exclusive,  of  course,  of  the  number  of  pieces  necessary  to  com- 
plete the  rows  when  the  corner  is  filled. 

Dividing  749  by  100,  the  number  required  to  make  a  row  on  each  of  the  two  sides, 
we  ascertain  that   about   7   additional 
3249  [  57  rows  can  be  made  out  of  the  remaining  107)749  {  7 

5  x  5  =  25  pieces.     But  we  can  not  complete  the  749 

107)749  square  by  adding  these  seven  rows,  un- 

749  less  we  can  add  also  seven  squares  to  each  row,  for  the 

corner  must  be  filled  in  order  that  we  may  have  a  square. 

Hence,  we  add  7  to  100  =  107,  and  divide  by  that  as  a  complete  divisor.  Uniting 
the  two  parts  of  the  arithmetical  operation,  we  have  the  work  on  the  left; 


SQUARES  AND   CUBES.  373 

Constructive  or  Geometric  Solution  of  the  Problem  of 
the  Cube  Root. 

Problem. — Let  it  be  required  to  find  the  edge  of  a  cube  which 
shall  contain  175616  cu.  inches. 

Solution. — Suppose  that  we  have  175616  cubic  blocks,  each  measuring  an  inch, 
and  let  it  be  required  to  make  of  them  one  cubic  block  as  large  as  possible. 

If  we  lay  down  100  blocks  in  one  row,  we  will  need  a  hundred  rows  for  the  first 
layer  of  our  large  cube.  100  in  a  row  and  100  rows  would  require  10000  blocks, 
and  the  hundred  layers  necessary  to  complete  the  work  would  take  1000000  blocks ; 
clearly,  then,  our  cube  can  not  ba  100  inches  in  length,  breadth,  and  thickness.  If 
we  take  90,  80,  70,  or  60  for  a  first  row,  and  try  to  complete  a  cube  of  so  many 
inches,  wc  shall  fall  short  of  blocks.  Such  repeated  trials  with  the  blocks,  how- 
ever, will  hardly  be  necessary.  A  few  trials  with  the  slate  and  pencil  would  lead 
the  beginner  to  discover  that  the  number  of  blocks  in  a  row  must  be  somewhere 
between  50  and  60,  and  hence  that  the  tens  in  the  root  can  not  exceed  5. 

To  build  up  a  cube  50  inches  in  length  will 
take  125000  blocks  (50  x  50  x  50).    How  many  175616  |  5» 

blocks  shall  we  have  left.     The  computation  in         50  X  50  X  50  =  125000 
the  margin,  which  will  be  readily  understood,  50616 

tells  us  that  we  shall  have  50616. 

In  making  additions  to  a  cube  for  the  purpose  in  view,  it  is  most  convenient  to 
make  them  to  three  sides. 

Remembering  that  the  block  already  constructed  measures  50  inches  long  and 
wide  and  high,  we  know  that  we  need  50  x  50  blocks  for  an  addition  of  one  layer 
to  one  side,  and  3  times  50  x  50  =  7500  for  an  addition  of  one  layer  to  each  of  the 
three  sides.  Can  we  add  7  such  layers  ?  Evidently  we  can  not,  for  7  times  7500 
blocks  would  be  52500.  Can  we  add  six?  Six  times  75C0  =  45000.  This  leaves 
us  50616  —  45000  =  5616  blocks. 

Will  5616  blocks  be  enough  to  complete  the  edges  and  the  corner?  To  test 
this  we  first  fill  the  upper  front  edge  by  placing  one  layer  after  another  on  the  top 
of  the  addition  already  made  to  that  side.     For  one 

layer  we  need  50  x  6,  and  for  six  layers  we  need  5616 

50  x  6  x  6,   or  1800  blocks ;  for  three  edges   we  3x50x6x6  =  5400 

need  3  times  50  x  6  x  6,  or  5400   blocks.     Taking  „-.« 

these  from  the  number  of  blocks  remaining,  we  have 
216  still  left  with  which  to  complete  the  work. 

Will  216  blocks  be  enough  to  fill  out  the  corner?  We  can  readily  see  that  we 
need  6  rows  of  6  blocks  each  for  one  layer  upon  the  upper  end  of  the  addition  made 
in  the  front  right-hand  edge,  and  that  we  must  have  6  such  layers  to  complete  the 
block;  6  x  6  x  6  =  21 6,  exactly  the  number  of  blocks  we  had  left. 

Thus  out  of  175616  small  cubic  blocks,  each  measuring  one  inch  in  length, 
breadth,  and  thickness,  we  have  constructed  one  large  block  measuring  50  +  6 
inches  on  each  edge.     Hence  56  is  the  cube  root  of  175616. 


374  STANDARD  ARITHMETIC. 

Applications  of  Square  and  Cube  Root. 

1.  5041  slabs  of  marble  9  inches  square  will  pave  a  square 
court.     How  many  slabs  on  each  side  ? 

2.  How  many  rods  long  is  the  side  of  a  square  field  containing 
10  acres  ?    40  acres  ?     90  acres  ?    490  acres  ?    640  acres  ? 

Note. — There  being  no  linear  unit  corresponding  to  the  acre,  acres  must  be 
reduced  to  other  denominations  before  we  attempt  the  solution  of  such  a  problem 
as  the  above. 

3.  Being  told  that  a  certain  cubic  block  of  marble  contained 
91  y8  cubic  feet,  I  measured  an  edge  and  found  it  to  be  4  ft.  5  % 
in.     How  much  more  should  it  have  measured  ? 

4.  What  must  be  the  inner  dimensions,  in  feet  and  inches,  of 
a  cubic  bin  containing  25,000  bushels  of  grain  ? 

5.  The  product  of  three  equal  numbers  is  3189506048.  What 
are  the  numbers  ? 

6.  A  rectangular  court  that  is  twice  as  long  as  it  is  wide  con- 
tains 31,250  square  feet.     How  long  and  wide  is  it  ? 

Suggestion. — If  the  rectangle  were  divided  into  two  equal 
squares,  how  many  square  feet  would  there  be  in  each  ? 


7.  A  block  of  stone  three  times  as  long  as  it  is  wide  and  high 
is  represented  to  contain  823  7/8  cubic  feet.     How  wide  and  high 

should  it  be  ?     (See  suggestion  under  Ex.  6.) 

8.  What  must  be  the  dimensions  in  feet  and  inches  of  a  square 
garden-lot,  which  shall  be  equal  to  two  rectangular  ones  measur- 
ing respectively  8  by  10  and  8  by  18  rods  ? 

9.  What  are  the  dimensions  of  a  cubic  bin  which  will  hold  as 
much  as  three  bins  measuring  respectively  12  by  18  by  9  ft.,  18  by 
27  by  6  ft.,  and  12  by  9  by  9  ft.  ? 

10.  A  bar  of  metal  5.75  ft.  long,  3.9  in.  wide,  and  .7  in.  thick 
being  melted  and  cast  into  cubic  form,  what  was  the  edge  of  the 
cube  ? 

11.  If  one  face  of  a  cube  contains  11  sq.  ft.  16  sq.  in.,  what  are 
the  contents  of  the  cube  ? 


SQUARES  AND  CUBES.  375 

Right-Angled   Triangles. 

380.  A  right-angled  triangle  is  a  triangle  that  has  one 
right  angle. 

381.  The  side  opposite  the  right  angle  is  called  the  hypote- 
nuse. 

Of  the  two  sides  forming  the  right 
angle,  either  one  may  be  taken  as  the 
base,  and  the  other  as  the  perpendicular. 

The  pupil  should  become  thoroughly  familiar  with  Base> 

the  following  proposition,  proved  in  geometry : 

382.  The  square  of  the  hypotenuse  is  equal  to  the  sum  of  the 
squares  of  the  other  two  sides. 

Example. — l.  The  base  of  a  right-angled  triangle  is  4  inches  ; 
the  perpendicular  is  3  inches.  What  is  the  length  of  the  hy- 
potenuse ? 

Solution. — The  square  of  the  base  is  9,  the  square 
of  the  perpendicular  is  16.     Hence,  to  obtain  the  square  3  2  =    9 

of  the  hypotenuse,  we  add  9  and  16  =  25,  the  square  48  =  16 

root  of  which  is  5  =  the  hypotenuse.  3  s  4-  4  2  =  25 

Note. — To  ascertain  whether  this  result  is  correct,  V25  __    5  hvD 

measure  the  distance  from  3  on  either  arm  of  a  carpen- 
ter's square  to  4  on  the  other. 

Example. — 2.  The  hypotenuse  of  a  right-angled  triangle  is  10 
inches  ;  the  perpendicular,  8  inches.     What  is  the  base  ? 

Solution. — The  square  of  the  hypotenuse  is  100, 

the  square  of  the  perpendicular  is  64.     To  obtain  the  1^2  -•  ™ 

square  of  the  base  we  subtract  64  from  100  (100  —  ft  2  _     _  . 

64  =  36).     The  sq.  **     ~    b4: 

root  of_36   is  the  10 2— 8^=    36 

base,  V36  =  6  Arts.         Base  =  V36  =  6  Ans. 

Note. — Test  cor- 
rectness of  the  answer  by  measuring  from  6  on  one 
N-.  arm  to  8  on  the  other  arm  of  a  carpenters'  square. 

I  1 1 » 1  1  1  *  1  1  1  rrT| 

383.  The  carpenters  square  is  an 
instrument  used  in  measuring  and  testing  the  work  of  the  car- 
penter, stone-mason,  etc. 


_3..      \  Note.— Test  cor- 

v\  rectness  of  the  answer  by  measuring  from  6  on  one 


Base. 

Perpen- 
dicular. 

Hypote- 
nuse. 

a.i%- 

% 

— 

4.  6 

4% 

— 

5.  2% 

3% 

'erpen- 

Hypote- 

icular. 

nuse. 

— 

n% 

T% 

12% 

11 



376  STANDARD  ARITHMETIC. 

EXERCISES. 

1.  If  the  width  of  a  book  is  9  inches,  and  the  length  12,  how 
many  inches  between  the  opposite  corners  ? 

2.  Two  dimensions  of  right-angled  triangles  being  given,  as 
follows,  find  the  third  dimension  of  each  : 

Base. 

6.  9 

7.  — 

8.  8% 

9.  If  a  room  is  21  ft.  long  and  20  ft.  wide,  how  long  is  the 
diagonal  ?  If  it  is  45  ft.  long  and  28  ft.  wide  ?  45  ft.  long  and 
24  ft.  wide  ?    24  ft.  long  and  10  ft.  wide  ? 

10.  Find  the  length  and  width  of  a  square  box  which  shall  con- 
tain as  much  as  two  boxes,  one  2  ft.  and  the  other  2  %  ft.  square, 
the  three  boxes  being  of  the  same  height.  If  one  is  4.2  and  the 
other  is  5.6  in.  square. 

11.  What  is  the  diameter  of  the  largest  circular  saw  that  can 
be  taken  through  a  doorway  8y2  ft.  highland  63/8  feet  wide  ?  If 
it  is  7%  ft.  high  and  5%  ft.  wide  ? 

12.  On  a  level  play-ground  there  is  a  rope,  11  y4  ft.  long,  fast- 
ened to  a  ring  at  the  top  of  a  pole  9  ft.  high.  How  far  from  the 
foot  of  the  pole  will  the  rope  reach  the  ground  ? 

13.  A  horse  is  to  be  tethered  in  the  center  of  a  rectangular  lot 
240  ft.  long  by  238  ft.  wide.  How  long  must  the  rope  be  which 
will  allow  him  to  graze  in  the  corners  of  the  lot  ? 

14.  The  figure  here  represents  a 
rectangular  farm.  The  only  dimen- 
sions given  are  1984  rods,  one  of  the 
longer  sides,  and  2434  rods,  the  di- 
agonal line.  How  many  acres  does 
the  farm  contain  ? 


CHAPTER    XIX. 


MENSURATION. 

I.    Plane   Surfaces. 

384.  If  a  straight-edge  laid  anywhere  upon  a  surface  touches 
at  every  point  the  surface  is  a  plane  surface. 

This  is  the  practical  test  of  a  plane.  It  can  be  tried  on  the  surface  of  a  desk, 
table,  floor,  wall,  or  any  other  surface  with  an  ordinary  straight-edge  rule.  The 
carpenter  uses  the  edge  of  his  plane  for  the  purpose. 

385.  A  portion  of  a  plane  bounded  by  one  or  more  lines  is  a 
plane  figure. 

386.  A  Circle  is  a  plane  figure  bounded  by  a 
curved  line,  every  point  of  which  is  equally  distant 
from  a  point  within,  called  the  center. 

387.  The  boundary  line  of  a  circle  is  called  the  circumfer- 
ence, n 

388.  A  straight  line  drawn  through  the  center 
and  terminating  at  the  circumference  on  both  sides 
is  called  a  diameter. 

389.  A  straight  line  drawn  from  the  center  to 

the  circumference  is  called  a  radius.     A  radius  is 

one  half  of  the  diameter.      (The  plural  of  radius  is  radii.) 

With  a  narrow  slip  of  paper  measure  the  circumference  of  a  dinner  plate,  then 
measure  the  distance  across  it,  and  you  will  find  the  circumference  a  little  more 
than  three  times  the  length  of  the  diameter.  If  the  plate  is  ten  inches  in  diameter, 
and  you  have  taken  the  measurements  carefully,  you  will  find  the  slip  of  paper  to 
be  very  nearly  2,\lj2  inches  long.  Thi3  result  agrees  very  nearly  with  the  great 
truth,  proved  in  geometry,  that 


378  STANDARD  ARITHMETIC. 

39  0«  The  circumference  of  any  circle  is  3.14159  times  the 
length  of  its  diameter.     Hence,  having  the  diameter, 

391.  To  find  the  circumference  of  a  circle:  Multiply  the 
diameter  by  3.14159. 

Conversely,  having  the  circumference, 

392.  To  find  the  diameter  of  a  circle:  Divide  the  circum- 
ference toy  3.14159. 

Note. — The  improper  fraction  corresponding  to  3.14159  may  be  remembered 
by  referring  to  the  series  of  figures,  113355  (the  first  three  digits  representing  odd 
numbers  written  doubly).  Taking  the  last  three  for  the  numerator  and  the  first 
three  for  the  denominator,  thus,  ff$f  we  have  the  ratio  of  the  circumference  to  the 
diameter.  The  reciprocal,  $$£,  represents  the  ratio  of  the  diameter  to  the  circum- 
ference. For  mere  approximations,  the  circumference  may  be  said  to  be  3  77 
times  the  diameter. 

SLATE     EXERCISES. 

1.  If  the  diameter  of  an  iron  column  is  3.5  in.,  what  is  the 
circumference  ?  If  the  girth  of  a  tree  is  5  ft.  9  in.  what  must  be 
its  diameter  ? 

2.  If  the  equatorial  diameter  of  the  earth  is  7925  miles,  how 
long  in  miles  and  rods  is  the  equator  ? 

3.  The  distance  from  the  center  of  the  hub  of  a  wheel  to  the 
outer  edge  of  the  felly  is  15  in.     How  long  must  the  tire  be  ? 

4.  If  the  length  of  an  oar  from  the  thole-pin  to  the  end  of  the 
blade  is  5  ft.,  how  many  feet  would  the  end  of  the  blade  travel 
in  the  water  during  6000  strokes,  each  describing  an  arc  of  60°  ? 

(60°  =  l/6  of  the  circumference.) 

5.  If  the  circumference  of  a  circular  pond  is  628.318  rods, 
what  part  of  a  mile  must  I  row  to  pass  from  shore  to  shore  across 
the  center  of  the  pond  ? 

6.  If  a  horse  is  tethered  to  the  middle  post  of  a  fence,  from 
which  he  can  graze  out  into  the  field  in  a  curved  line  78.539314 
ft.  long,  how  long  is  the  tether  ? 

7.  What  will  be  the  circumference  of  the  largest  circle  that 
can  be  drawn  on  a  sheet  of  paper  12  in.  wide  and  18  in.  long  ? 


MENSURATION. 


379 


393.  A  plane  figure  bounded  by  three  straight  lines  is  a 
triangle. 

394.  The  base  of  a  triangle  is  the  side  on  which  it  is  sup- 
posed to  rest.      (Any  side  of  a  triangle  may  be  taken  for  its  base.) 

395.  The   altitude  of  a  triangle  is  the  perpendicular  dis- 
tance from  the  angle  opposite 
the  base,  to  the  base,  or  to  the 

base    produced.       {Produced— con- 
tinued in  the  same  direction.) 

396.  Triangles  take  different  names,  according  to  the  rela- 
tions of  their  sides.  If  the  sides  of  a  triangle  are  equal,  it  is  an 
eqilateral  triangle.  If  only  two  sides  of  a  triangle  are  equal, 
it  is  an  isosceles  triangle.  If  no  two  sides  are  equal,  it  is  a 
scalene  triangle.  If  one  of  the  angles  is  a  right  angle,  it  is  a 
right-angled  triangle,  or  a  right  triangle. 


Equihteral 
Triangle. 


Scalene 
Triangle. 


Eight-angled 
Triangles. 


397.  Parallel  lines  are  straight  lines  that  have  the  same 
direction  but  do  not  coincide,  and  can  never  meet,  however  far 
they  may  be  produced. 

398.  A  plane  figure  bounded  by  four  straight  lines  is  a  quad- 
rilateral.   (  Quadrilateral  means  four-sided.) 

399.  Quadrilaterals  take  different  names  from  their  angles 
and  from  the  relation  of  the  sides  to  each  other. 

400.  A  quadrilateral  which  has  no  two  sides 
parallel  is  a  trapezium. 


401.   A  quadrilateral   which  has  only  two 
sides  parallel  is  a  trapezoid. 


380 


STANDARD  ARITHMETIC. 


4-02.  A  quadrilateral  that  has  its  opposite  sides  parallel  is  a 
parallelogram. 

403.  Parallelograms    take   different    names  from   their 
angles  and  the  relation  of  the  sides  to  each  other. 


404.  A   parallelogram   that  has   all   its  angles 
right  angles,  and  all  its  sides  equal,  is  a  square. 

405.  A  parallelogram  that  has  all  its 
angles  right  angles,  and  only  its  opposite  sides 
equal,  is  called  a  rectangle. 

406.  A  parallelogram  that  has  its  sides  all 
equal,  but  whose  angles  are  not  right  angles,  is 
called  a  rhombus. 

407.  A  parallelogram  that  has  only  its 
opposite  sides  equal,  and  whose  angles  are  not 
right  angles,  is  called  a  rhomboid.  , 


408.  A  straight  line  that  joins  the  vertices  of  two  angles, 
not  adjacent,  is  a  diagonal. 

To  find  the  Areas  of  Quadrilaterals  and  Triangles. 

The  Rectangle,  including  the  Square. —We  have 
already  found  that  'to  compute  the  area  of  a  rectangle,  we  must 
multiply  the  number  of  the  proposed  square  units  of  measure 
which  can  be  placed  on  one  side  of  the  rectangle  by  the  number 
of  corresponding  linear  units  in  the  adjacent  side. 

Example. — Let  the  figure  represent     cu  .,,.,.,.,,,,,,,,,,,,£ 
a  rectangle    14  inches   wide   and   19 
inches  long.     What  is  the  area  ?  * 

Solution. — 19  square  inches  can  be  placed 
on  the  side  c  d,  and,  since  there  are  14  such 
rows,  there  will  be  14  times  19  sq.  in.  in  the 
whole  rectangle  =  266  sq.  in. 


MENSURATION.  381 

The  Rhomboid  and  Rhombus.— Example.— Let  it  be  re- 
quired to  compute  the  area  of  a  rhomboid,  10  in.  long  and  6  in.  wide. 

Solution. — 6  x  10  □  in.  =  60  □  in.  Am. 

If  from  either  end  of  a  rhomboid  we  cut  a  right-angled 
triangle,  and  add  it  to  the  other  end,  as  indicated  by  dotted 
lines  in  the  figure,  we  should  form  a  rectangle  equivalent  to 
the  rhomboid ;  hence,  '  ' 

4-09.  To  find  the  area  of  a  rhomboid :  Multiply  the  length 
of  one  of  two  parallel  sides  by  the  distance  between  them. 

The  rule  for  the  rhombus  is  the  same.  , -~« 

It  is  to  be  observed  that,  to  obtain  the  width  of  a  rhombus  /  j  ^^/ 
or  rhomboid,  we  do  not  measure  a  side,  but  the  perpendicular  ,• L^^  / 
distance  between  parallel  sides.  ^— I ' 

The  Triangle— Example.— Given  the  base  of  a  triangle  14 
yd.  and  the  altitude  9  yd.,  to  find  the  area. 

The  triangle  is  one  half  of  a  parallelogram  having  the  same  base  and  altitude, 
as  may  be  seen  by  the  above  diagram.     Hence, 

4-10.  To  find  the  area  of  a  triangle:  Find  the  area  of  a 
rectangle  of  the  same  base  and  altitude,  and  take  one  half  of  it. 

411.  The  following  rule  is  sometimes  necessary  : 

When  the  three  sides  of  a  triangle  are  given,  to  find  the 
area :  From  half  the  sum  of  the  three  sides  subtract  each  side 
separately.  Multiply  the  half  sum  and  the  three  remainders  to- 
gether, and  extract  the  square  root  of  the  product. 

The  Trapezoid. —Example.  — Given    the         > 

length  of  each  of  the  two  parallel  sides  of   a  /  jV 

trapezoid,  6  and  10  feet,  and  the  distance  be-  T i  A 

tween  them,  5  feet,  to  find  the  area. 

Solution.— 6  ft,  +  10  ft.  =  16  ft.  x/2  of  16  ft.  =  8  ft.  5x8  sq.  ft.  =  40 
sq.  ft.  Ans. 

By  inspection  of  the  figure  it  will  be  seen  that,  by  the  aid  of  dotted  lines,  we 
have  constructed  a  rhomboid  equal  in  area  to  the  trapezoid  whose  area  is  required ; 
and,  further,  it  is  plain  that  the  side  of  this  rhomboid  is  equal  to  half  the  sum  of 
the  two  parallel  sides  of  the  trapezoid.     Hence, 

4-12.    To  find  the  area  of  a  trapezoid:  Multiply  one  half 
the  sum  of  the  parallel  sides  by  the  distance  between  them. 
17 


382  STANDARD  ARITHMETIC. 

4-13.  Th.8  Trapezium. — The  surface  of  a  trapezium  may  be 
found  by  dividing  it  into  two  triangles  ;  then  having  measured 
the  length  of  the  diagonal  and  the  two  perpendiculars,  we  calcu- 
late the  area  of  each  triangle  separately.  The 
sum  of  the  areas  of  the  two  triangles  is  the 
area  of  the  trapezium. 

Example. — In  a  trapezium  A  B  C  D,  we 
measure  the  diagonal  A  C,  and  find  it  to  be 
24  feet ;  also  the  perpendiculars,  and  find 
one  to  be  18,  the  other  9  feet.  What  is  the 
area  of  the  trapezium  ? 

Solution. 
Area  of  the  triangle  A  B  C  =  24  x     9  =  216  sq.  ft.,  l/9  of  216  =  108  sq.  ft. 
Area  of  the  triangle  A  D  C  =  24  x  18  =  432  sq  ft.,  lfy  of  432  =  216  sq.  ft. 
Area  of  A  B  C +A  D  C,orthewho\etm.=  648  sq.  ft.,  1/i  =1*24  sq.  ft.  Am. 


SLATE      EXERCISES. 


l.  How  many  acres  in  a  piece  of  woodland  220  yd.  in  length 


and  1  furlong  in  width  ? 

2.  How  many  square  miles  in  a  township  5  miles  and  40  chains 
square  ? 

3.  How  many  square  feet  in  a  floor  20  ft.  long  and  5  yd. 
wide  ? 

4.  Find  the  surface  of  a  pane  of  glass  measuring  37%  in. 
long  and  23  in.  wide. 

5.  How  many  square  yards  in  the  four  walls  of  a  room  15  ft. 
6  in.  high  and  80  ft.  in  compass  ? 

6.  A  rectangular  pavement,  50  ft.  9  in.  long  and  12  ft.  6  in. 
wide,  was  laid  with  a  central  line  of  stone  5  ft.  wride  at  $1.75  a 
running  foot ;  the  sides  were  flanked  with  brick  at  80^  per  square 
yard.     What  did  the  paving  cost  ? 

7.  How  many  square  feet  in  a  surface  24  ft.  long  20  ft.  wide  ? 
How  many  in  another  surface  of  half  these  dimensions  ? 


MENSURATION.  383 

8.  I  have  a  box  without  a  lid  ;  it  is  5  ft.  long,  4  ft,  wide,  and 
3  ft.  deep,  interior  dimensions.  How  many  square  feet  of  zinc 
will  it  take  to  line  the  bottom  and  sides  of  this  box  ? 

9.  Find  the  area  of  a  rhomboid  whose  length  is  1  yd.  2  ft. 
6  in.,  and  whose  width  is  2  ft.  3  in.  Draw  this  figure  on  your 
slate,  with  the  scale  reduced  by  12. 

10.  What  is  the  height  of  a  rhomboid  whose  area  is  12  A.  and 
its  length  13y3  chains? 

11.  The  four  eaves  of  a  pyramidal  roof  measure  each  44  ft.  3 
in.,  and  the  common  peak  of  the  four  triangles  has  a  perpendicu- 
lar distance  of  24  ft.  from  the  eaves.  What  is  the  area  in  slaters' 
squares  (1)  of  one  triangle  ?  (2)  /of  the  roof  ? 

12.  I  have  a  triangular  garden  containing  233  %  square  yards. 
The  perpendicular  distance  from  the  apex  to  the  base  is  20  ft. 
What  is  the  length  of  the  base  ? 

13.  A  triangular  field,  whose  sides  are  unequal,  contains  5 
acres.  The  base-line  measures  y4  mile.  What  is  the  altitude  in 
chains  ? 

14.  What  is  the  area  of  a  triangle  whose  three  side3  are  13,  14, 
and  15  ft.? 

15.  What  is  the  area  in  acres  of  a  triangular  field  whose  three 
sides  measure  respectively  47,  58,  and  69  rods  ? 

16.  The  parallel  sides  of  a  trapezoid  measure  respectively  3y3 
ft.  and  6  ft.  8  in.  ;  the  perpendicular  distance  between  them  is 
2  ft.     What  is  the  area  ? 

17.  Find  the  area  of  a  trapezium  whose  diagonal  is  168,  and 
one  perpendicular  42,  the  other  56. 

18.  How  many  centares  in  a  rhomboid  one  side  of  which  meas- 
ures 50  meters,  the  perpendicular  distance  to  the  opposite  side 
being  35  meters  ? 

19.  What  is  the  area  of  a  square  field,  the  diagonal  of  which 
measures  174  meters  ? 


384 


STANDARD  ARITHMETIC. 


Regular  Polygons,  having  more  than  four  sides, 

take  different  names  according  to  the  number  of  sides.     The 


following  are  some  of  them  : 


PeDtagon. 


Heptagon 


Octagon. 


Nonagon.  Decagon. 

414.  By  dividing  a  regular  polygon  into  triangles  by  lines 
drawn  from  the  center  to  the  several  angles, 

it  will  be  readily  seen  that  the  area  of  the 
polygon  is  equal  to  the  sum  of  the  areas  of 
the  triangles ;  hence, 

415.  To  find  the  area  of  a  regular  poly- 
gon :  Multiply  the  perimeter  (the  sum  of  all 
the  sides)  by  one  half  the  perpendicular  dis- 
tance from  the  center  to  one  of  the  sides. 

Example. — The  side  of  a  pentagon  measures  5  ft.,  and  the 
perpendicular  distance  from  the  center  to  the  side  is  4%  ft. 
What  is  the  area  of  the  polygon  ? 

Solution. — The  area  of  each  triangle  =;  1/2  (4T/2  x  5),  that  is,  one  half  the 
product  of  the  base  by  the  altitude,  and  the  area  of  the  whole  polygon  =  1/2  (4  l/2  x 
25)  =  56  y4   □  ft.  Ans. 

416.  The  Circle* — The  calculation  of  the  areas  of  polygons 
leads  us  by  an  easy  step  to  the  calculation  of 
the  area  of  a  circle,  for  it  is  plain  that,  as  the 
number  of  sides  of  the  polygon  is  increased, 
the  perimeter  becomes  more  and  more  nearly 
equal  to  the  circumference  of  the  circum- 
scribed circle,  and  the  perpendicular  distance 
from  the  center  to  the  sides  of  the  polygon 
becomes  more  and  more  nearly  equal  to  the  radius  of  the  circle. 
Hence,  having  the  circumference  and  the  radius, 

417.  To  find  the  area  of  a  circle :  Multiply  the  circumfer- 
ence by  one  half  of  the  radius. 


MENSURATION. 


385 


Another  Method  of  finding  the  Area  of  a  Circle. — If  we 
have  a  square,  and  find  the  center  of  it  by  lines  joining  op- 
posite corners,  and  with  this  center  inscribe  a  circumference 
exactly  touching  the  sides  of  the  square,  the  corners  outside 
of  the  circle  will  contain  .2146,  and  the  circle  itself  .7854  of 
the  surface  of  the  square.     Hence,  we  have  another  rule, 

4-18.  To  find  the  surface  of  a  circle:  Mul- 
tiply the  square  of  the  diameter  by  .7854. 


S.ATE      EXERCISES. 

1.  Required  the  area  of  a  regular  hexagon  whose  side  is  73  ft. 
and  the  perpendicular  is  63.2  ft. 

2.  How  many  acres  in  an  octagonal  section  of  land  whose 
side  is  1983.2  rods  and  perpendicular  7%  miles  ? 

3.  How  many  square  yards  are  there  in  a  circle  whose  diame- 
ter is  12  ft.   6  in.  ? 

4.  Draw  a  square  containing  81  square  inches ;  inscribe  a 
circle  in  this  square.  What  is  the  superficies  of  this  circle  in 
square  inches  ? 

5.  A  cow  is  tethered  to  a  post  driven  in  the  center  of  a  lot 
100  ft.  square ;  the  tether  is  just  long  enough  for  her  to  reach 
the  fence.  How  much  of  the  surface  of  the  field  is  she  unable 
to  crop  ? 

6.  Find  the  radius  of  a  circle  whose  area  is  95.0334  square  ft. 

7.  Find  the  area  of  a  tent  floor,  with 

/  A  7? 

semicircular  ends,  from  the  dimensions  of 
the  following  diagram,  in  which  the  line 
A  B  equals  200  ft.  and  the  line  A  G 
equals  90  ft. 

8.  What  is  the  difference  in  length  between  a  fence  around  a 
circular  lot  123  ft.  in  diameter  and  a  square  lot  of  the  same  width  ? 

9.  Find  the  difference  in  cost  at  87 1/2<f  per  rod  between  fenc- 
ing a  square  field  of  10  acres  and  a  rectangular  field  32  rods 
wide  of  the  same  area. 


OZZD 


386  STANDARD  ARITHMETIC. 

II.    Mensuration  of  Solids. 

419.  A  Solid  is  a  limited  portion  of  matter  haying  length, 
breadth,  and  thickness. 

420.  A  solid  bounded  by  a  curved  surface, 
every  point  of  which  is  equally  distant  from  a 
point  within,  called  the  center,  is  called  a 
globe,  or  sphere. 

A  circle  which  divides  the  surface  of  a 
sphere  into  two  equal  parts  is  called  a  great 
circle  of  the  sphere.      \  Sphere. 

421.  The  surface  of  a  sphere  is  exactly  equal  to  four  times 
the  surface  of  a  great  circle  of  the  sphere. 

It  would  require  just  four  times  as  much  gold-leaf  to  cover  a 

sphere  as  would  cover  one  side  of  the  section  made  by  cutting  the 

sphere  into  two  equal  parts  (hemispheres). 

It  will  be  recollected  that  the  surface  of  a  circle  is  found  by  multiplying  the 
circumference  by  one  half  the  radius ;  hence,  to  find  the  surface  of  the  sphere,  we 
multiply  its  circumference  by  four  times  one  half  the  radius,  or,  in  other  words,  by 
the  diameter.     Hence, 

422.  To  find  the  surface  of  a  sphere:  Multiply  the  cir- 
cumference by  the  diameter. 

Conversely,  having  the  surface, 

423.  To  find  the  diameter  of  a  sphere:  Divide  the  surface 
of  the  sphere  by  4  to  obtain  the  surface  of  a  great  circle.  Divide 
this  by  .7854  to  find  the  area  of  the  circumscribed  square.  Ex- 
tract the  square  root  to  find  the  side  of  the  square.  The  side  of 
the  circumscribed  square  is  equal  to  the  diameter  of  the  circle. 


Applications. — l.  If  the  earth  were  a  sphere  with  a  diameter 
of  7925  miles,  what  would  its  whole  surface  be  ? 

2.  How  many  square  inches  of  leather  would  it  require  to  cover 
a  ball  3  in.  in  diameter  ? 

3.  The  surface  of  a  sphere  is  11170.12  square  feet.     What  is 
its  diameter  ? 


MENSURATION. 


387 


4-24-.  Prisms. — A  prism  is  a  solid  whose  bases  are  equal  and 
parallel  polygons,  and  whose  faces  are  parallelograms. 


t-25.  Prisms  take  different  names  from  the  forms  of 
their  bases.  It  will  be  seen  that  cubes  and  rectangular  solids 
are  prisms.     They  are  also  called  Parallelopipeds. 


A,  a,  a,  a,  are  bases;  b,  b,  b,  b,  are  lateral  faces;  the  edges  between  the 
faces  are  lateral  edges  (side  edges) ;  the  edges  between  the  faces  and  bases  are 
basal  edges  (base  edges).  If  the  faces  and  edges  arc  perpendicular  to  the  bases, 
the  prism  is  said  to  be  a  right  prism,  otherwise  it  is  said  to  be  oblique. 


426.  The  sum  of  all  the  lateral  faces  is  the  convex  surface. 
The  sum  of  bases  and  faces  is  the  entire  surface.  The  altitude 
is  the  shortest  distance  between  the  bases. 

Since  the  faces  of  prisms  are  all  parallelograms,  having  a  common  altitude,  if 
we  have  a  side  of  the  base  and  the  altitude  given, 

4-27.  To  find  the  convex  surface  of  a  prirm :  Multiply  the 
perimeter  (sum  of  all  the  sides)  of  the  base  by  the  altitude  of  the 
prism. 

To  find  the  entire  surface :  Add  the  areas  of  the  bases  to 
the  convex  surface. 

Suggestion.  —  Suppose  that  you 
have  a  block  of  the  shape  of  one  of 
these  prisms,  and  that  you  have  fitted 
to  it  a  piece  of  paper  so  as  to  exactly 
cover  its  convex  surface.  The  paper 
will  be  a  parallelogram,  one  side  being 
equal  to  the  height  and  the  other  side 
to  the  perimeter  of  the  base.  Hence 
the  rule  as  riven  above. 


■388 


STANDARD  ARITHMETIC. 


428.   Cylinders.— When  the  number  of 

sides  of  the  bases  of  a  right  prism  is  so  increased 

that  the  bases  become  circles,  the  prism  becomes 

a  cylinder. 

In  this  case  the  prism  loses  the  lateral  edges,  and  the 
faces  become  one.  Other  terms  used  for  prisms  apply  also 
to  the  corresponding  dimensions  of  the  cylinder. 


Cylinder. 


4-29,  To  find  the  convex  surface  of  a 
cylinder :  Multiply  the  circumference  of  the  base  by  the  altitude 
of  the  cylinder. 

4-30.  To  find  the  entire  surface:  Add  the  areas  of  the  bases 
to  the  convex  surface. 

431.  Pyramids. — A  solid  that  has  a  polygon  for  its  base, 
and  triangles,  meeting  in  a  point,  for  its  faces,  is  a  pyramid. 

432.  The  vertex  is  the  point  in  which  the 
triangular  faces  meet.  The  altitude  is  the 
shortest  distance  from  the  vertex  to  the  base. 
The  slant  height  is  the  shortest  distance  from 
the  vertex  to  one  of  the  sides  of  the  base.  Other 
names  applied  to  the  parts  of  the  prism  apply 
also  to  the  corresponding  parts  of  the  pyramid.  pyramid. 

433.  Since  the  faces  of  a  pyramid  are  equal  triangles  having 
the  sides  of  the  base  for  their  bases,  and  the  slant  height  for  their 
common  altitude, 

434.  To  find  the  convex  surface  of  a  pyramid :  Multiply 
the  perimeter  of  the  base  by  one  half  the  slant  height. 


Applications. — l.    Find   the  entire   surface   of  an   octagonal 
prism,  12  in.  high  with  2  in.  sides. 

2.  Find  the  convex  surface  of  a  piece  of  stove-pipe,  6  in.  in 
diameter  and  2  ft.  in  length. 

3.  Find  the  convex  surface  of  a  great  pyramid,  764  feet  square, 
and  having  a  slant  height  of  451  feet. 


MENSURATION. 


389 


4-35.  The  Cone.— When  the  number  of 
the  sides  of  the  base  of  a  pyramid  is  so  increased 
that  the  base  becomes  a  circle,  the  pyramid  be- 
comes a  cone. 

From  the  explanation  of  the  rule  for  finding  the  convex 
surface  of  a  pyramid,  the  reason  of  the  following  rule  is 
plain.  Having  the  circumference  of  the  base  and  the  slant 
height  given, 

436.  To  find  the  convex  surface  of  a  cone:  Multiply  the 
circumference  of  the  base  by  one  half  the  slant  height. 

437.  Frustums  of  Pyramids  and  Cones.— If  a  part  of 
a  pyramid  or  cone  be  cut 
away,  so  that  the  section  is 
parallel  to  the  base,  the  por- 
tion between  the  section  and 
the  base  is  called  the  frus- 

Frustum  of  a  pyramid.      turn  of  the  pyramid  or  cone. 


Frustum  of  a  cone. 


A  face  of  a  frustum  of  a  pyramid  is  evidently  a  trapezoid,  the  surface  of 
which  is  found  by  multiplying  one  half  the  sum  of  the  parallel  sides  by  the  slant 
height  of  the  frustum.     Hence,  having  a  side  each  of  the  upper  and  lower  bases, 

4-38.  To  find  the  convex  surface  of  a  frustum  of  a  pyra- 
mid: Multiply  half  the  sum  of  the  perimeters  of  the  upper  and 
lower  bases  by  the  slant  height. 

Also,  having  the  circumference  of  the  upper  and  lower  bases, 

4-39.  To  find  the  convex  surface  of  a  frustum  of  a  cone  : 

Multiply  half  the  sum  of   the  circumferences  of  the  upper  and 
lower  bases  by  the  slant  height. 


Applications. — l.  What  is  the  convex  surface  of  the  frustum  of 
a  hexagonal  pyramid,  each  side  of  the  lower  base  being  4  ft.,  and 
of  the  upper  base  3  ft.,  and  the  slant  height  8  ft.  ? 

2.  What  is  the  convex  surface  of  the  frustum  of  a  cone,  the 
diameter  of  the  lower  base  being  18  in.,  that  of  the  lower  base 
8  in.,  and  the  slant  height  8y3  in.? 


390 


STANDARD  ARITHMETIC. 


The  Volume  or  Contents  of  Solids. 
Contents  of  Prisms  and  Cylinders.— To  find  the  volume  or  contents  of 

rectangular  solids,  we  have  learned  to  multiply  the  number  of  solid  units  which  can 
be  laid  upon  the  base  by  the  number  of  layers  necessary  to  complete  a  parallelo- 
piped  of  the  required  height ;  or,  as  it  is  commonly  expressed,  we  multiply  the 
base  by  the  altitude.  The  rule  is  the  same  for  all  prisms,  and  also  for  cylinders. 
Hence, 

4-4-0.  To  find  the  contents  of  prisms  and  cylinders :  Hav- 
ing found  the  base  of  the  prism  or  cylinder  by  the  rules  for  plain 
surfaces,  multiply  the  base  by  the  altitude.  The  product  will  be 
the  volume  sought. 

Contents  of  Pyramids  and  Cones.— It  is  proved  in 

geometry  that  the  volume  of  a  pyramid  is  exactly  one  third 
of  that  of  a  prism  which  has  the  same  base  and  altitude,  and 
that  the  volume  of  a  cone  is  exactly  one  third  of  that  of  a 
cylinder  having  the  same  base  and  altitude. 

If  a  solid  iron  cylinder  of  any  dimensions  were  turned 
down  to  a  cone,  as  represented  in  the  cut,  the  cone  would 
weigh  just  one  half  as  much  as  the  parts  cut  away,  that  is, 
only  one  third  the  solid  contents  of  the  cylinder  would  remain; 
hence  we  have  the  rule : 

4-4-1.  To  find  the  contents  of  pyramids  and  cones:  Hav- 
ing found  the  base,  multiply  it  by  the  altitude  of  the  pyramid  or 
cone,  take  one  third  of  the  product,  and  the  result  will  be  the 
contents  required. 

Contents  Of  the  Sphere.— As  the  mode  of  cal- 
culating the  area  of  a  triangle  was  applied  to  the  cal- 
culation of  the  area  of  a  circle,  so  the  mode  of  finding 
the  contents  of  a  pyramid  may  be  applied  to  the  calcu- 
lation of  the  contents  of  the  sphere. 

For,  as  the  triangles  into  which  a  polygon  can  be 
divided  may  be  regarded  as  being  so  increased  in  num- 
ber that  their  bases  finally  become  the  circumference  of 
a  circumscribed  circle,  so  the  number  of  faces  of  a  solid 
similar  to  the  one  here  represented  may  be  regarded  as 
being  so  increased  that  they  become  the  faces  of  a  solid  differing  so  little  from  a 
perfect  sphere  that  the  solid  may  be  regarded  as  a  sphere  composed  of  a  great  num. 
ber  of  pyramids,  all  the  bases  of  which  make  up  the  surface  of  the  sphere ;  hence, 

4-4-2.  To  find  the  contents  of  a  sphere:  Multiply  the  sur- 
face of  the  sphere  by  one  third  of  the  radius. 


MENSURATION. 


391 


Another  Method  for  finding  the  Solid  Contents  of  a  Sphere. — The  solid  con- 
tents of  any  sphere  are  .5236  of  a  cube  whose  edges  meas- 
ure the  same  as  the  diameter  of  the  sphere.  (A  base-ball 
that  just  touches  every  side  of  the  box  containing  it  occu- 
pies a  little  more  than  one  half,  or  more  nearly  .5236,  of  the 
space  in  the  box.)     Hence,  having  the  diameter, 

4-4-3.  To  find  the  solid  contents  of  a 
sphere :  Find  the  contents  of  a  cube  whose 
edges  are  equal  to  the  diameter,  and  take  .5236 
of  the  result. 


Applications. — l.  What  is  the  solidity  of  a  triangular  prism 
whose  length  is  12  ft.,  and  either  of  the  equal  sides  of  one  of  its 
equilateral  ends  is  3  ft.  ? 

2.  How  many  gallons  of  water  would  a  cylindrical  boiler  con- 
tain if  25  in.  high  and  12  in.  in  diameter  ? 

3.  Find  the  cubic  inches  in  the  largest  cone  that  can  be  cut 
from  a  cylinder  2  ft.  6  in.  high  and  14  in.  in  diameter. 

4.  A  sphere  8  in.  in  diameter  is  placed  in  a  cubic  box  whose 
interior  dimensions  are  8  in.     How  much  vacant  space  is  left  ? 

5.  I  have  a  cylindrical  tank  which  contains  160  gallons  ;  it  is 
6  ft.  5  in.  in  diameter.     How  deep  is  it  ? 

6.  Find  the  cubic  feet  in  a  log  30  ft.  long  and  2  ft.  in 
diameter  at  the  larger  and  1  ft.  10  in.  at  the  smaller  end. 

7.  Find  the  cubic  contents  of  the  great  pyramid  mentioned 
in  Problem  3,  page  388. 

8.  How  many  cubic  feet  in  a  circular  mound  48  ft.  high,  and 
having  a  diameter  of  86  ft.  at  the  top,  and  a  circumference  of 
471.24  ft.  at  the  bottom? 

9.  How  many  cubic  miles  in  the  earth,  supposing  it  to  be  a 
perfect  sphere  8000  miles  in  diameter  ? 

10.  How  many  barrels  of  oil  in  a  tank  60  ft.  in  diameter  if 

the  Oil  is  5   ft.  deep  ?      (40  gal.  to  the  barrel.) 

11.  Find  how  many  cubic  meters  in  a  sphere,  the  surface  of 
which  contains  5682  a  meters/ 


392 


STANDARD  ARITHMETIC. 


Duodecimals. 

444.  A  scale  of  tivelfths,   called   a   Duodecimal  Scale,  is 
sometimes  used  in  the  measurement  of  surfaces  and  solids. 

Example. — What  is  the  area  of  a  table  top  which  is  3  ft.  9  in. 
long  by  2  ft.  7  In.  wide  ? 

Ft.       12ths.  Solution. — Write  one  dimension  under  the  other,  calling 

2  7  ■     the  inches  12ths  (of  a  foot).     Then  multiplying  by  3,  we  have 

g  c>  3  x  7/12  =  21/i2  =  1  sq.  ft.  and  9/12  sq.  ft.  =  the  area  of 

the  narrow  strip  at  the  bottom  and  on  the  left  of  the  arrow. 
Writing  the 

3.         12ths  under 

9.  8.        3  I2ths,     and 

adding  the 
units  to  the  product  of  3  x  2  (area 
of  the  large  squares  above)  we  have 
3  x  2  +  1  =  7  sq.  ft.  Writing  the 
7  under  ft.,  we  have  7  sq.  ft.  and 
9  twelfths  of  a  sq.  ft.,  the  entire 
area  of  that  part  of  the  figure  on 
the  left  of  the  arrow. 


9 

11 


Next,  9/i2  x  7i 


/144  — 


3  feet 

kt//Z 

,  *i 

k\ 

1 

5  twelfths  and  3/i44,  the  area  of 

the  small  rectangle   at  the  lower  H 

right-hand  corner.     Writing  the  3 

in  the  line  below  the  first  partial  product,  and  to  the  right,  as  a  lower  denomination, 

and  adding  the  5  twelfths  to  9/i  2  x  2  (the  area  of  the  remaining  part  of  the  figure), 

we  have  23/12  =  1  and  ll/i2.     Writing  the  X1/i2  under  the  12ths,  and  the  1  under 

the  ft.,  we  have  in  the  second  part  of  the  product  the  entire  area  of  that  part  of 

the  figure  which  is  represented  on  the  right  of  the  arrow.     The  sum  of  the  two 

parts  thus  found  is  the  entire  area  required. 

445.  In  duodecimals,  the  unit  is  a  linear  foot,  a  square  foot, 
or  a  cubic  foot,  according  as  it  is  used  to  represent  the  length  of  a 
line,  the  area  of  a  surface,  or  the  contents  of  a  solid. 

One  twefth  of  a  foot  is  a  prime  ('),  t/-i2  of  a  prime  is  a 
second  ("),  1/12  of  a  second  is  a  third  ('"),  etc. 

Hence,  for  linear,  square,  and  cubic  measure,  wc  need  only  the  following 

Tabla : 
12 ""  =  1 '"         12  '  m  1 "         12    =  1  12  =  1  foot 


MENSURATION.  393 

Addition  and  subtraction  of  duodecimals  are  performed  as  in 
compound  numbers.    For  multiplication,  we  observe  the  following 

446.  Rule. — 1.  "Write  the  terms  of  the  multiplier  under  the 
corresponding  terms  of  the  multiplicand. 

2.  Multiply  each  term  in  the  multiplicand,  beginning  with  the 
lowest,  by  the  highest  term  of  the  multiplier,  then  by  the  next 
lower,  etc.,  observing  that  12  of  any  lower  denomination  make 
one  of  the  next  higher. 

3.  Add  together  the  partial  products  thus  obtained. 


35  ft. 

9' 

10  ft. 

6' 

350  ft. 

90' 

210' 

54" 

SLATE      EXERCISES. 

l.  Find  the  area  of  a  rectangle  measuring  35  ft.  9'  by  10  ft.  6\ 

Solution. 

35  fL       9'  Or,  all    reductions 

10  fo»       6  may   be  made   in   the 

.   357  ft.       6'  process  of  adding  the 

17  ft.     10*     6"  partial  products,  as  at 

m~K       V~¥'         the  riSht-  375  ft.         4'      6" 

A  t  14$  per  u  yard,  find  the  cost  of  painting 

2.  64  ft.  3'  by  25  ft.  3'  4.  108  ft.  9'  by  31  ft.  6' 

3.  28  ft.  6'  by  17  ft.  9'  5.  36  ft.  8'  by  38  ft.  8' 

At  16$  a  □  yard :  At  the  prices  given  per  □  yard : 

6.  65  ft.  9'  by  1  ft.  1'  6"  9.  198  ft.  9'  by  2  ft.  V  at  100 

7.  34  ft.  10'  6"  by  2  ft.  3'  10.  283  ft.  9'  by  3  ft.  4'  at  110 

8.  123  ft.  4'  6"  by  1  ft.  8'  3"  11.  114  ft.  6'  by  5  ft.  11'  at  120 

Find  the  solid  contents  of  blocks  of  marble  measuring 

12.  3  ft.  2'  by  2  ft.  V  by  1  ft.  6'        15.  7  ft.  2'  by  4  ft.  5'  by  3  ft.  6' 

13.  4  ft.  9'  by  1  ft.  9'  by  2  ft.  3'        16,  10  ft.  1'  by  3  ft.  2'  by  4  ft.  3' 

14.  5  ft.  3'  by  5  ft.  2'  by  15  ft.  9'      17.  7  ft.  8'  by  8  ft.  7'  by  6  ft.  5' 

447.  The  process  of  division  being  the  reverse  of  multiplica- 
tion, no  rule  is  needed. 

10  ft.  6' )  375  ft.     4'  6"  (35  ft.  9' 
Example.— Divide  375  ft.  367  ft.    6' 

4'  6"  by  10  ft.  6'.  7  ft.  10'  6" 

7  ft.  10'  6" 


394  STANDARD  ARITHMETIC. 

Original  Problems. 

1.  Drop  a  plumb-line  from  a  window  ;  mark  the  distance  from 
the  ground,  and  find  what  else  is  needed  to  make  a  problem  re- 
quiring the  distance  from  the  sill  of  the  window  to  any  distant 
object  on  the  ground. 

2.  The  masts  used  for  electric  lights  in  many  cities  will  sug- 
gest good  problems. 

3.  Several  pieces  of  smooth,  straight  wire  of  uniform  length, 
being  bent  into  outlines  of  various  geometric  planes,  will  suggest 
many  problems.  How  should  one  of  these  wires  be  bent  to  inclose 
the  greatest  space,  in  the  shape  of  a  triangle,  of  a  square,  of  a 

circle,   or  what  ?     (A  good  foot-rule  should  be  used  to  take  the  necessary 
dimensions.) 

4.  Card-boards  being  placed  in  their  hands,  the  members  of 
the  class  may  be  asked  to  make  boxes  of  them  so  that  there  shall 
be  only  one  piece  in  a  box ;  to  make  the  convex  surfaces  of  cylin- 
ders, cones,  prisms,  pyramids,  etc.  Ask  how  a  circular  card-board 
may  be  cut  so  that,  when  the  cut  edges  are  neatly  joined  edge  to 
edge,  the  surface  of  the  cone  thus  produced  may  contain  %  or 
other  specified  fractional  part  of  the  surface  of  the  circle. 

5.  Ask  the  class  to  take  the  necessary  measurements  and  to 
calculate  the  contents  of  a  block  of  stone,  of  a  large  water-pipe, 
of  a  log  of  wood,  or  other  suitable  object  in  the  neighborhood. 

6.  Ask  the  diameter  and  solid  contents  of  the  largest  cylinder, 
the  largest  cone,  or  of  the  largest  sphere  that  can  be  cut  from  a 
rectangular  block  of  wood  which  you  may  bring  into  the  class- 
room, leaving  the  members  of  the  class  to  make  their  own  meas- 
ments. 

7.  Construct  a  cubic  box  into  which  a  sphere  may  be  placed 
touching  all  the  sides,  and  take  the  ball  out  and  fill  the  box  with 
sand  ;  weigh  that,  and  ask  the  class  what  the  weight  of  the  sand 
would  be  if  it  had  been  filled  in  around  the  ball.  Weigh,  and  see 
how  nearly  the  two  results  correspond. 


CHAPTER    XX. 

EXCHANGE.— DUTIES   OR   CUSTOMS.— BONDS. 

448.  Exchange  is  a  method  by  which  one  person  may  make 
a  payment  to  another  person  at  a  distance  without  transmitting 
money. 

Illustrations. — 1.  A  familiar  illustration  of  exchange  is  found  in  the  use  of 
Postal-Notes  and  Money-Orders  for  paying  small  sums  to  persons  at  a  distance. 

2.  If  larger  ones  are  to  be  paid,  the  Government  leaves  the  business  to  private 
persons,  usually  bankers,  who  have  credit  and  the  necessary  understanding  with 
each  other.  Thus,  if  a  person  in  New  York  buys  merchandise  of  another  at  New 
Orleans,  he  does  not  send  gold  or  bank-notes  to  pay  for  it,  but  goes  to  a  New  Yo:k 
bank  and  buys  an  order,  called  a  draft,  on  a  bank  in  New  Orleans,  for  the  amount 
desired.  This  he  sends  to  his  creditor,  who  takes  it  to  the  banker  on  whom  it  is 
drawn  and  gets  the  money  for  it. 

3.  Postal  notes  and  money-orders  are  always  payable  at  sight,  that  is,  when  pre- 
sented, but,  on  such  orders  as  those  just  mentioned,  time  is  often  allowed  for  pay- 
ment. Hence,  the  banker  in  New  York  who  gets  "cash  down"  for  a  draft  allows 
interest  on  the  sum  till  the  time  set  for  its  payment  in  New  Orleans. 

4.  So  long  as  New  York  is  buying  cotton  and  sugar  from  New  Orleans,  and  New 
Orleans  is  buying  manufactured  goods  from  New  York  to  about  equal  amounts,  the 
sums  to  be  paid  in  each  city  by  the  merchants  of  the  other  are  about  equal,  and  the 
number  of  drafts  drawn  in  each  city  on  the  other  is  about  the  same ;  but,  if  New 
York  were  buying  $10,000,000  worth  per  month  and  New  Orleans  only  $5,000,000 
worth,  money  would  have  to  be  sent  from  New  York  to  New  Orleans  to  pay  the 
balance.  In  this  case,  the  banks  of  New  York,  in  selling  drafts  on  New  Orleans, 
would  charge  something  to  pay  for  the  risk  and  expense  of  shipping  money  to  New 
Orleans ;  and,  on  the  other  hand,  the  banks  of  New  Orleans  would  be  glad  to  sell 
drafts  on  New  York  for  something  less  than  their  face,  for  thus  they  would  be 
getting  some  of  the  balance  due  them. 

5.  The  rates  of  premium  on  drafts  for  considerable  sums  can  be  but  little 
greater  than  the  charges  made  by  the  express  companies  for  carrying  the  money, 
for,  if  a  merchant  in  Chicago  had  to  pay  $10,075  for  a  draft  of  $10,000  on  New 
York,  and  the  charge  for  expressage  were  but  $50,  he  would  save  $25  by  sending 
the  money  directly.     •-,-••• 


396  STANDARD  ARITHMETIC. 

Definitions. 

4-49.  A  Draft  is  a  written  order  directing  or  requesting  the 
party  to  whom  it  is  addressed  to  pay  a  certain  sum  to  a  certain 
person,  or  to  his  order,  and  to  charge  the  same  to  the  person  who 
makes  the  request. 

450.  Drafts  payable  in  the  country  in  which  they  are  drawn 
are  called  Domestic  or  Inland  Bills;  when  payable  in  a  foreign 
country,  they  are  most  commonly  called  Bills  of  Exchange. 

451.  The  party  making  the  order  is  the  Drawer;  the  party 
ordered  or  requested  to  pay  is  the  Drawee ;  the  party  named  to 
whom  or  to  whose  order  the  payment  is  to  be  made  is  the  Payee. 

452.  The  payee  may  transfer  a  draft  to  another  party  in  the 
same  way  that  he  would  transfer  a  promissory  note  made  payable 

to  his  order.      (See  page  321,  note  2.) 

453.  A  Sight  Draft  is  payable  when  presented.  A  Time 
Draft  is  payable  at  the  time  named  in  the  draft. 

Three  days'  grace  are  allowed  on  time  drafts  and  sometimes  on  sight  drafts. 

4-54.  The  following  is  the  common  form  of  domestic  bills  : 
Sight   Draft. 

$500.  Cleveland,  0.,  May  15,  1886. 

At  sight,  pay  to  the  order  of  Henry  James  &  Sons,  five 
hundred  dollars,  and  charge  to  the  account  of 

M.  W.  Chester. 
To  Wilson  &  Hewitt,  Boston,  Mass. 

Notes. — 1.  If  time  is  allowed  for  the  payment  of  a  draft,  it  should  be  made  to 
read  Thirty  days  after  sight,  Sixty  days  after  sight,  etc.,  according  to  the  time  agreed 
upon.     In  this  case  it  would  be  called  a  Time  Draft. 

2.  When  a  time  draft  is  received,  the  payee  should  at  once  present  it  to  the 
drawee,  who  writes  the  word  Accepted,  with  the  amount,  date,  and  his  own  name, 
across  the  face  of  the  draft,  if  he  is  willing  to  honor  it,  that  is,  pay  the  sum  called 
for.     The  time  in  which  it  matures  is  then  reckoned  from  the  date  of  accevt&nce. 


EXCHANGE.  397 

SLATEEXERCISES. 

On  May  14,  1886,  exchange  on  New  York  was  quoted  as  below  in  the  several 
cities  named : 

Chicago,  50  premium.         Charleston,    */t  and  lf4ft  premium. 

St.  Louis,  25  and  50         "  New  Orleans,  100         " 

Savannah,    3/16  and  l/A%         "  San  Francisco,    15  and  20         " 

Boston,  17  and  20  premium. 

Note. — When  exchange  is  quoted  at  a  given  sum,  it  means  so  much  on  $1000. 
50^  on  $1000  is  equivalent  to  1/2o^ 

1.  Find  the  cost  in  Milwaukee  of  a  sight  draft  on  New  York 
for  $600,  exchange  being  %  fo  premium. 

Analysis. — Since  the  draft  is  sold  at  a  premium  of  3/4  $,  each  dollar  costs 
$1.00  3/4,  and  $600  costs  600  times  1.00  3/4  =$604.50  Ans. 

2.  How  much  must  be  paid  in  Chicago  for  a  sight  draft  on 
New  York  for  $11,200,  the  discount  being  3/16$  ? 

Analysis. — Since  the  discount  on  the  face  of  the  draft  is  3/i6  #,  each  dollar 
costs  $.9913/16,  and  $11,200  costs  11,200  times  $.9913/16  =$11,179  Ans. 

What  must  be  paid  for  a  draft  drawn  at 

3.  New  York  on  Cleveland  for  $1500  at  */,£  discount? 


4. 

Detroit 

"   San  Francisco    " 

$500 

"   1  <f0  premium  ? 

5. 

Cincinnati 

w   Louisville           " 

$1725 

"    1U$  discount? 

6. 

Charleston,  S.  C, 

"  New  York          * 

$850 

"   $2.50  discount? 

7. 

Savannah 

"   Chicago              " 

$1600 

"  $1.25  premium? 

8. 

Philadelphia 

"  New  Orleans      " 

$2500 

"  $1.50  discount? 

9.  A  dealer  of  New  York  buys  500  barrels  of  pork  in  Chicago 
at  $12.50  per  barrel,  and  pays  by  draft  at  30  days  after  date. 
What  does  the  draft  cost  him,  exchange  being  50^  discount  ? 

Solution.— The  interest  on  $6250  for  33  days  at  6%  is  $34,375,  and  the  dis- 
count for  exchange  (^20^  of  $6250)  is  $3,125;  hence  the  total  discount  on  the 
face  of  the  bill  =  $37.50.  This  being  deducted  from  $6250,  the  cost  of  the  draft 
is  found  to  be  $6212.50. 

Or,  Discount  on  $1  for  33  days    =  .0055 

Exchange  discount          x\iq%  =  .0005 
Total  discount  on  $1  .006 

Hence  $1   of  exchange  costs  $.994,  and  $6250  costs  6250  times  $.994  =  $6212.50. 


398  STANDARD  ARITHMETIC. 

What  must  be  paid  for  a  draft  drawn  at 

10.  St.  Louis  on  Atlanta  for  $525,  payable  60  d.  after  date,  at  1l2fo 
discount  ? 

11.  St.  Augustine  on  Philadelphia  for  $400,  payable  90  d.  after  date, 
at  1/8^  premium? 

12.  New  Orleans  on  Buffalo  for  $1950,  payable  30  d.  after  date,  at 
1l'8</o  discount? 

13.  Philadelphia  on  Baltimore  for  $275,  payable  60  d.  after  date,  at 
4/5$  discount? 

14.  New  York  on  Charleston,  S.  C,  for  $3000,  payable  90  d.  after  date, 
at  $1.25  premium? 

15.  What  is  the  face  of  a  sight  draft  bought  for  $7500  at  a 
premiumof  $2.50  ?    ($2.50  on  $1000=  l/^%.) 

Analysis. — At  a  premium  of  1U%,  $1  exchange  can  be  bought  for  $1.00  74, 
and  as  many  dollars  of  exchange  can  be  bought  for  $7500  as  $1.00  1/4  is  contained 
times  in  $7500.  $7500  -5-  1.00  x/4  =  $7481.30;  hence  $7500  will  purchase  a  draft 
for  $7481.30. 

What  is  the  face  of  a  sight  draft  bought  for 

16.  $1000,  premium  being  \x\%<f>.         21.  $050,  discount  being  1/2</c. 

17.  $250,           "            "      lfo.               22.  $1225,  premium  "      3/4$. 

18.  $4300,  discount       "      llU</o.        .23.  $360,  discount  "      2/5<f0. 

19.  $2900,         "             u      $1.75.           24.  $500,  premium  "      $2. 

20.  $2625,  premium      "      $1.25.           25.  $3115,        "  "      $2.25. 

26.  Find  the  largest  draft  payable  30  days  after  date  that  can 

be  bought  for  $4985.00,  exchange  being  at  a  premium  of  y4#. 

Solution. — The  cost  of  $1  exchange,  on  the  con-  -. 

ditions  given  above,  is  found  as  already  shown.    From  ' 

$1,  the  interest  for  83   days  at  6^  =  $.0055  is  dc- 
ducted,  and  to  the  remainder  the  premium  at  ll±%  =  .9945 

.0025  is  added.     The  sum  is  the  cost  of  $1  exchange.  .0025 

Hence,  as  many  dollars  exchange  can  be  bought  for  ,997 

$4985  as  $.997  is  contained  times  in  $4985  =  5000. 
Hence  a  draft  for  $5000  can  be  bought  for  $4985.  $4985  -f-  $997  =  5000 

Note. — The  result  of  the  above  process  is  exact,  but  business  men  would  greatly 
shorten  the  work  by  adding  3/x  0  of  1  %  to  $4985,  and  thus  obtain  a  result  correct  to 
within  41/z^>  which  would  be  considered  sufficiently  accurate.  The  answers  given 
to  the  following  examples  arc  found  by  the  process  given  in  the  solution. 


EXCHANGE.  399 

SLATE      EXERCISES. 

Interest  being   6%,  find   the  face  of  a   30-day  draft  that  can  be 
bought  for 

27.  $500,  ex,  being  1/B^  premium.        31.  $1216,  ex.  being  3/4$  premium. 

28.  $325,   "       "      */4#  discount.         32.  $1925,    "       "      */8#  discount. 

29.  $90,      "        u      lf0  "  33.  $2500,    "       "      V8^Premium- 

30.  $720,   u       "      i/g^  premium.        34.  $1650,    «       "      8/5  $  discount. 

/U  £/?e  same  rate  of  interest,  find  the  face  of  a  60-day  draft  that 
can  be  bought  for 

35.  $375,  ex.  being  7/8$  discount.  40.  $750,  ex.  being  5/6$  premium. 

36.  $1465, "       "      i/*#        "  41.  $2000,  "       M      3/5  %  discount. 

37.  $1390, "       "      8/8^Premium-  42.  $5650,  "       "      %fa         M 

38.  $1500, "       "      9/10^  discount.       43.  $560,    "       "      8/4$  Premhmi- 

39.  $695,    "       "      1/s#Premium-  44.  $225,    "       "      1$  discount. 

The  rate  of  interest  being  4%,  find   the  face   of  a  90-day  draft 
that  can  be  bought  for 

45.  $2600,  ex.  being  */8#  discount.  43.  $5000,  ex.  being  3/5  %  discount. 

46.  $700,      "       "      */*#  premium.  49.  $3750,    "       "      B/s#  premium. 

47.  $1950,    "       «      1V8^       "  50.  $9000,    "       M      l'/^dis't. 


Applications. — l.  A  merchant  in  St.  Louis  ordered  his  broker 
in  New  York  to  purchase  $5000  worth  of  merchandise  for  him. 
On  shipping  the  goods,  the  broker  draws  on  the  merchant  for  the 
amount,  with  commission  at  3  $.  What  should  be  the  face  of 
the  draft,  the  premium  on  St.  Louis  being  1/2  <f0  ? 

2.  Find  the  face  of  a  60-day  draft,  bought  for  $620.75,  if  ex- 
change is  $2.50  discount,  and  interest  6$. 

3.  A  merchant  in  Chicago  pays  $1075  in  Kansas  City  by  a  30- 
day  draft.     What  does  the  draft  cost,  exchange  being  4/5$  prem.  ? 

4.  What  per  cent,  of  its  face  is  the  cost  of  a  90-day  draft,  if 
exchange  is  1  $  premium,  and  interest  is  allowed  at  4$  ? 

5.  If  exchange  is  $2.50  premium  and  interest  6$,  what  will  be 
the  cost  in  Philadelphia  of  a  draft  on  New  Orleans  for  $1600, 
payable  60  days  after  date  ? 


400 


STANDARD  ARITHMETIC. 


Foreign  Exchange^ 

4-55.  Foreign  bills  of  exchange  are  drafts  drawn  in  one  coun- 
try  and  payable  in  another. 

Note. — Bills  drawn  in  one  State,  and  payable  in  another,  are  termed  foreign 
bills  in  the  laws  of  some  of  the  States. 

Foreign   Bill  of  Exchange. 


Sixty  days __after. iMiM... of  this  First  of  Exchange, 
Second  and  Third  of  the  same  tenor  and  date  unpaid, 

pay  to  the  order  of Dftvid  Mason _^  Co., 

Six  Hundred  Fifty-six  Pounds  Sterling, 

Value  received,  and  charge  the  same  to  the  account  of 
To .Q^jypoV^^T^^.^.CO'  .GJJiV3iQVP.iM9PA^Cp- , . 


London. 


In  making  foreign  bills,  it  is  customary  to  draw  three  of  the  same  tenor  and 
date,  called  a  set  of  exchange,  each  of  which  contains  a  clause  rendering  all  of 
them  worthless  except  the  one  first  presented  for  payment.  These  bills,  or  two  of 
them,  are  sent  by  different  mails  to  avoid  the  inconvenience  of  delay  if  one  should 
be  lost.     The  third  is  sometimes  retained  by  the  buyer  or  remitter. 

456.  The  Par  of  Exchange  between  two  countries  is  the 
value  of  the  standard  coins  of  one  country  in  the  standard  coins 
of  the  other. 

Thus,  the  par  of  exchange  between  the  United  States  and  England  is  $4.8665; 
that  is,  the  value  of  the  gold  in  the  sovereign  of  Great  Britain  is  found  by  careful 
analysis  to  be  worth  $4.8665  in  the  gold  of  United  States  currency.  (For  par  of 
exchange  between  the  United  States  and  other  foreign  countries,  see  page  415.) 

4-57.  In  consequence  of  fluctuations  in  demand  for  bills  of 
exchange  (as  explained  in  §4,  Art.  448),  they  are  commonly  at  a 
slight  premium  or  discount,  that  is,  a  little  above  or  below  par. 


EXCHANGE.  401 

The  following  were  the  rates  posted  in  the  offices  of  dealers  in  foreign  exchange 
in  New  York  city,  May  14,  188G  : 

60  days.  3  days. 

Sterling 4.87  7*  4-t,° 

Paris  (francs) 5.15  5/8  5.12  »/, 

Hamburg  (reichsmarks). 95  3/4  .96  3/8 

Berlin  (reichsmarks) 95  7 /8  .96  3/8 

Amsterdam  (guilders) 40  7s.  .40  3/4 

Quotations  for  3  days  refer  to  sight  exchange,  on  the  theory  that  3  days'  grace 
are  allowed  on  sight  drafts,  though  custom  varies  in  this  respect.  The  reason  for 
charging  less  for  time  drafts  than  for  sight  bills  is,  that  the  banker  who  sells  them 
has  the  use  of  the  money  from  the  time  the  draft  is  drawn  till  it  is  paid,  as  already 
explained.  Sight  drafts  are  sometimes  called  "short"  exchange,  and  sixty-day 
drafts  "long"  exchange. 

1.  At  the  rates  quoted  above,  find  the  cost  of  a  draft  on  Lon- 
don for  £1896  10.  & 

£1896  10.   6  ==  £1896.525 

1896.525  X  $4. 87  7s  =  $9245.56- 

Explanation. — Reducing  10s.  6c?.  to  the  decimal  of  a  pound  sterling,  we  find 
£1896  10.  6  to  be  equal  to  £1896.525,  and,  since  the  cost  of  £1  is  $4.87  72,  the 
cost  of  £1896.525  will  be  1896.525  times  $4.87  l/s  =  $9245.56—  Am. 

2.  Find  the  cost  in  New  York  of  a  draft  on  Paris  for  80,568.25 
francs,  at  the  rate  quoted. 

80,568.25  fr.  -J-  5.13*/g  fr.  =  $1572.06+ 

Explanation. — Since  5.12  72  francs  =  $1,  as  many  dollars  must  be  paid  for 
the  draft  as  there  are  times  5.12  ll2  francs  in  80,568.25  francs  =e  $1572.06+  Am. 

At  the  rates  quoted,  find  the  cost  of  a  sight  draft  in  New  York  on 

3.  London    for  £200.  6.  Paris  for     500  fr. 

4.  Paris  H     1000  fr.  7.  Amsterdam  "      720  guil. 

5.  Hamburg  t;       300  marks.  8.  Berlin  "     2300  marks. 

9.  Find  the  face  of  a  bill  on  London  that  can  be  bought  in 

New  York  for  $793,  the  rate  of  exchange  being  $4.90. 

$793  -r-  $4.90  =  £161.8367+ 

£161.8267+ =  £161  16  9- 

Explanation.— Since  $4.90  will  buy  £1  of  exchange,  $793  will  buy  as  many 
pounds  exchange  as  $4.90  is  contained  times  in  $793  =  £161   16  9—  Am. 


402  STANDARD  ARITHMETIC. 

10.  Find  the  face  of  a  3-day  (sight)  bill  on  Hamburg,  bought 
in  New  York  for  $100,000,  exchange  being  963/8. 
$100,000  -f-  96%  =  103,761.35 
103,761.35  X  4  marks  =  415,045.4  marks. 

Explanation.— Since  4  marks  can  be  bought  for  $.963/8,  as  many  times  4 
marks  can  be  bought  for  $100,000  as  $.963/8  is  contained  times  in  $100,000  = 
103,761.35  at  415,045.4  marks  Am. 

At  the  same  rates,  find  the  face  of  a  60-day  draft,  bought  in  New 
York,  on 

11.  Paris      for  $5000.  14.  Antwerp  for    $900. 

12.  London   "   $6500.  15.  Paris  "    $2175. 

13.  Berlin      "    $8000.  16.  Hamburg    "    $1900. 


Applications.— l.  A  merchant  wishes  to  send  $2200  to  his  cor- 
respondent in  Paris.  Find  the  face  of  a  sight  draft,  exchange 
being  5.15y2. 

2.  Find  the  cost  of  a  bill  of  exchange  on  Berlin  for  2500  marks, 

the  rate  being  .96.      (The  quotation  is  for  4  marks.) 

3.  Find  the  face  of  a  sight  draft  on  London,  purchased  in 
Boston  for  $5222.75,  exchange  at  4.87%. 

4.  A  merchant  purchased  a  sight  draft  on  London  for  £625 
10.  6.     Exchange  being  4.90,  what  did  he  pay  for  it  ? 

5.  Find  the  value  of  a  60-day  draft  on  Paris  for  5000  francs, 
exchange  being  5. 14  %. 

6.  I  paid  $2575  for  a  draft  on  Hamburg.  How  many  marks 
did  the  draft  call  for,  exchange  being  .95%  ? 

7.  If  I  paid  $580.80  for  a  draft  of  £120  on  London,  what  was 
the  rate  of  exchange  ? 

8.  A  hardware  merchant  in  Louisville  purchased  cutlery  in 
Sheffield,  England,  the  bill  for  which  was  £785  15.  What  was 
the  cost  of  the  draft  sent  in  payment,  exchange  being  4.82  ? 

9.  How  much  must  be  paid  for  a  bill  of  exchange  on  Amster- 
dam for  15,640  guilders  at  3  days'  sight,  exchange  being  40%  and 
brokerage  %$  ? 


DUTIES  OR  CUSTOMS.  403 

Duties  or  Customs. 

458.  Duties  or  customs  are  taxes  upon  imported  goods. 

Notes. — 1.  Places  designated  by  government  for  the  collection  of  duties  are 
called  Ports  of  Entry.  Each  port  of  entry  has  a  Custom  House,  which  is  in 
charge  of  the  Collector  of  Customs  for  that  district. 

2.  A  tax  called  tonnage  is  levied  upon  vessels  according  to  the  number  of 
tons  they  are  estimated  to  carry.  The  collection  of  this  tax  belongs  to  the  customs 
officers. 

459.  A  duty  fixed  at  a  certain  per  cent,  on  the  cost  of  an 
imported  article  is  called  an  ad  valorem  duty  (i.  e.,  duty  based 
on  value  or  cost). 

The  cost  on  which  an  ad  valorem  duty  is  calculated  is  the  net  cost  of  the  mer- 
chandise in  the  country  from  which  it  is  imported,  as  ascertained  from  an  invoice, 
which  is  required  to  be  exhibited  by  the  importer  when  he  applies  for  a  permit  to 
land  his  goods.  This  invoice  or  manifest  must  be  accompanied  by  a  consular  cer- 
tificate that  the  prices  of  the  goods  given  in  the  manifest  are  the  prices  that  prevail 
in  the  market  where  they  were  purchased.  This  precaution  is  taken  to  prevent 
undervaluation,  whereby  the  revenues  are  sometimes  defrauded. 

460.  A  duty  assessed  on  each  article,  or  on  each  pound,  yard, 
etc.,  of  imported  goods,  is  called  a  specific  duty. 

On  some  goods,  both  specific  and  ad  valorem  duties  are  required  to  be  paid. 
Thus,  a  specific  duty  of  35^  per  lb.,  and  an  ad  valorem  duty  of  40  per  cent.,  are 
both  assessed  on  all  imported  woolen  knit-goods. 

In  assessing  specific  duties,  the  collector  must  make  allowance  "  for  the  differ- 
ence between  the  invoiced  and  ascertained  quantity,"  duties  being  exacted  on  "  the 
quantity  of  merchandise  which  arrives  in  the  United  States,  not  on  the  quantity 
shipped  at  the  foreign  port." — (General  Regulations  under  the  Customs  and  Navi- 
gation Laws,  Art.  604.) 

^_ 

SLATE     EXERCISES. 

1.  Find  the  duty  on  5000  lb.  raisins  at  2f  per  lb. 

2.  Find  the  duty  paid  on  5300  boxes  of  oranges,  invoiced  at 
15s.  per  box,  at  20  %  ad  valorem. 

3.  What  is  the  duty  on  1000  yards  of  carpet,  invoiced  at  4s. 
per  yard,  at  300  per  yard  and  30  f0  ad  valorem. 

4.  At  5<fi  per  yard  and  35  $  ad  valorem,  what  is  the  duty  on 
500  yards  of  dress-goods  invoiced  at  20^  per  yard  ? 


404  STANDARD  ARITHMETIC. 

5.  At  $2  per  ton,  what  is  the  duty  on  7565  lb.  of  hay  ? 

6.  Find  the  duty  paid  on  a  shipment  of  knit-goods  weighing 
623  lb.,  and  valued  at  $2340,  at  35^'  per  lb.  and  40$  ad  valorem. 

7.  Find  the  duties  paid  on  the  following  importation  :  500 
boxes  of  oranges  at  $4  per  box,  at  20  $  ad  valorem  ;  700  lb.  raisins 
at  2<f>  per  lb. ;  1000  lb.  figs  at  2<f  per  lb. ;  750  boxes  lemons  at  30# 
per  box  ;  and  $250  worth  of  preserved  fruit  at  35  $  ad  valorem. 

8.  What  was  the  duty  paid  on  12  pianos,  valued  at  $200  each, 
at  25  $  ad  valorem  ? 

9.  The  duty  on  steel  bars  being  45$  ad  valorem,  what  was  the 
duty  paid  on  8500  lb.,  invoiced  at  4^  per  lb.? 

10.  A  lady,  on  returning  from  Europe,  brings  with  her  laces 
and  insertings  for  which  she  had  paid  350  marks ;  gloves,  for 
which  she  had  paid  60  francs  ;  a  picture,  for  which  she  had  paid 
1200  francs ;  and  three  uiused  ready-made  dresses,  which  cost 
her  780,  930,  and  800  francs.  How  much  duty  did  she  have  to 
pay,  the  duty  on  laces  and  insertings  being  30$,  gloves  50$, 
pictures  30$,  and  ready-made  clothing  50$  ?    (See  Table,  page  415  ) 


Bonds. 

461.  A  bond  is  a  written  instrument  given  to  secure  the  dis- 
charge of  an  obligation. 

Bonds  issued  by  a  government  or  corporation  to  secure  the  payment  of  money 
correspond  to  promissory  notes  issued  by  individuals.  They  are  made  payable  at  a 
certain  time,  and  bear  a  specified  rate  of  interest.  The  interest  on  bonds  is  com- 
monly made  payable  annually,  semi-annually,  or  quarterly. 

Note. — An  individual  who  wishes  to  borrow  money  for  his  own  use,  can  do  so 
from  any  one  who  will  lend  it  to  him.  If  he  pays  more  interest  than  he  needs  to, 
it  is  his  own  loss.  But  one  who  borrows  for  others  must  borrow  at  the  lowest  pos- 
sible rates  of  interest,  or  else  there  is  good  ground  for  complaint. 

Hence,  when  the  United  States  Government,  a  State,  county,  city,  or  incorpor- 
ated company,  finds  it  necessary  to  borrow  money,  bonds  are  prepared,  and  these  are 
sold  to  the  highest  bidders ;  that  is,  those  who  desire  to  loan  the  money,  and  will 
pay  the  highest  premium  for  the  privilege  of  doing  it.  This  effects  the  same  result 
as  if  the  bonds  were  offered  to  those  who  would  lend  the  money  at  the  lowest  rates 
of  interest. 


BONDS.  405 

4-62.  Registered  bonds  are  recorded,  with  the  names  of  their 
owners,  and  can  not  be  transferred  from  one  party  to  another 
without  a  change  of  the  record. 

4-63.  Coupon  bonds  are  bonds  to  which  certificates  are  at- 
tached calling  for  the  payment  of  certain  interest  at  specified 
times. 

These  certificates  called  coupons  are  cut  off  and  presented  for  payment  when 
they  become  due.  No  record  is  made  of  the  holders  of  coupon  bonds,  hence  they 
may  be  transferred  from  one  person  to  another  by  delivery  as  bank-notes. 

464.  Bonds  issued  by  the  United  States  Government  (some- 
times called  Government  securities),  and  State  bonds  (called  State 
securities),  are  distinguished  by  their  rates  of  interest,  dates  at 
which  they  are  made  payable,  etc. 

Thus,  in  the  daily  papers  of  May  20,  1886,  we  find  mentioned  "  IT.  S.  41/g,s»  '91, 
reg.,"  that  is,  United  States  registered  bonds  bearing  VJ2%  interest  and  payable  in 
1891.     (See  quotations,  Art.  465.) 

465.  Bonds,  like  stocks,  are  bought  and  sold  at  the  Stock 
Exchanges  of  all  the  principal  cities,  the  brokerage  on  the  pur- 
chase and  sale  of  both  being  the  same.     (See  Art.  291,  page  291.) 

The  following  are  the  quotations  for  the  United  States  securities,  in  the  market 
May  22,  1886  : 

Bid.      Asked.  Bid.      Asked. 

U.  S.  New  3 1005/8     —  U.  S.  Currency  6*8,  1895..  1275/8     — 

U.S. 47g,  1891,  regist'd.  Ill l/8  lll3/8  "  1896..  1301/,,     — 

U.S.472,  1891.  coupon.  1123/8  11272  "  1897..  1323/8     — 

U.  S.  4,  1907,  registered.  1253/4  12578  "  1898..  13578     — 

U.  S.  4,  1907,  coupon  ...  125  3/4  125  7/8  "  1899..  1378/8     — 

Note. — Currency  6's  are  bonds  issued  by  the  United  States  Government  in 
aid  of  the  trans-continental  railways,  bearing  6  %  interest,  and  payable  in  currency 
at  the  times  specified  in  the  quotations. 


SLATE      EXERCISES. 
In  the  following  problems,  $100  bonds  are  referred  to,  and  brokerage  is  reck- 
oned at  1/s%  on  par  values,  unless  otherwise  stated. 

1.  Find  the  cost  to  the  buyer  of  38  bonds  at  97  %. 

2.  If  a  person  sells  38  bonds  at  97  %,  what  will  he  receive 
from  his  broker  ? 

18 


406  STANDARD  ARITHMETIC. 

3.  What  amount  in  bonds    at    112 1/2    can    be    bought    for 
$10,586.75  ? 

The  brokerage  being  added  to  the  price  of  the  bond,  the  cost  of  1  bond  is  found 
to  be  112  5/8.     At  this  rate,  how  many  can  be  bought  for  the  given  sum  ? 

Note. — No  principles  are  involved  in  the  solution  of  problems  like  the  fore- 
going but  such  as  are  applicable  to  the  purchase  and  sale  of  stocks ;  but  when 
questions  arise  as  to  the  advantage  of  investing  in  one  kind  of  bonds  rather  than 
another,  such  problems  as  the  following  occur : 

4.  If  a  person  buys  5  f0  stock  at  125,  what  rate  of  interest 
does  he  receive  on  the  money  invested  ? 

Whatever  he  pays  for  the  bond,  the  interest  he  receives  is  always  the  interest 
specified  in  the  bond.  Hence  he  receives  $5  interest  on  his  investment.  What  per 
cent,  of  $125  is  $5  ? 

Solution. 

5  -125  =  5/i25  =  7*5  =4%  /m 

Or,  if  stock  at  100  yields  5$,  at  125  it  must  yield  100/125,  or  */5  of  5f0  =4%. 

What  %  interest  will  be  realized  on  money  invested 

5.  In  4$       bond3  at     80.  9.  In  8f0        bonds  at  160. 

6.  In  Qfo  "  "  120.  10.  In  5  */,  +  "  "  110. 
VlMVajf  "  "  90-  11.  In  6V4#  "  "125. 
8.  In  33/4$     "      M     75.                  12.  In  8f0  "      "   140. 

How  much  money  must  be  invested 

13.  In  4$      bonds  at     92  3/8  to  produce  $352  income. 

14.  In  3$  "    .  "   100  78    "        "         $540 

15.  In  **/i*     "      "   108         "        «         $630        " 

16.  In  4  7a$     "      "   108         "        u        $630        " 

17.  That  I  may  receive  6  $  on  the  money  invested,  what  price 
may  I  pay  for  8  $  bonds  ? 

What  would  be  the  difference  between  the  income  from  an  invest- 
ment 

18.  Of  $8400  in  4*/f»i  at  120,  and  in  3V2's  at  112. 

19.  Of  $2700  "   6's         "   135,     "     "  4V2's   "     90. 

20.  Of  $4500  "   3V2's  «     90,     "      u   3's         "     75. 

21.  If  a  person  invest  $4488  in  3  fo  stocks  at  70,  and  $5505  in 
currency  6's  at  137 1/2,  paying  the  usual  brokerage,  what  percent, 
will  his  income  be  on  the  sum  invested  ? 


APPENDIX. 


Testing  the  Accuracy  of  Addition,  Subtraction,  Multi- 
plication, and  Division. 

466.  There  is  no  better  way  of  making  sure  of  the  correct- 
ness of  an  addition  than  adding  "both  ways";  and  in  subtrac- 
tion the  best  test  of  accuracy  is  that  the  sum  of  the  subtrahend 
and  remainder  is  equal  to  the  minuend. 

In  multiplication,  the  most  thorough  proof  of  accuracy  is  found  if  the  product 
of  the  multiplier  by  the  multiplicand  is  equal  to  the  first  product.  In  division,  if  the 
sum  of  the  remainder  added  to  the  product  of  the  quotient  by  the  divisor  is  equal 
to  the  dividend,  the  work  may  be  relied  upon  as  correct.  But  these  methods  of 
proof  require  as  much  time  as  the  original  operation,  hence  the  common  use  of  the 
method  called 

Casting  out  Nines, 

To  cast  out  the  nines  of  a  number  we  may  add  all  the  terms 
of  the  number,  and  divide  the  sum  by  9 ;  the  remainder  will  be 
:he  result  sought.  Thus,  the  sum  of  the  terms  of  4787763  is  42. 
Dividing  this  by  9,  we  obtain  6  as  the  excess  of  nines.  But, 
since  we  wish  to  know  only  what  the  remainder  is,  we  may  drop 
the  nines  from  the  results  as  we  proceed. 

Thus,  in  the  operation  of  casting  out  nines  from  4*787763,  we  may  think  the 
process  indicated  in  light-faced  italics,  and  speak  the  numbers  printed  in  full-faced 
type,  as  follows : 

4  +  7  =  11,  11  -  9  =  2;  2  +  8  =  10,  10  -  9  =  1;  1  +  7  +  7  =  15,  15  - 
9  =  6  ;  6  +  6  =  12,  12  —  9  =  3  ;  3  +  3  =  6.  6  being  the  excess  of  nines,  is 
written  as  it  is  pronounced. 

In  this  process  we  ship  the  9's,  for  there  is  no  use  of  adding  a 
nine  and  at  once  dropping  it. 


408 


STANDARD  ARITHMETIC. 


Written  Work. 

3885— 6 

647—5 


27195  48—3 
15540 
23310 
2513595—3 


Applied  to  Multiplication. 

Explanation. 

Remainder  after  casting  out  nines  from  3885. 

647. 
Multiplying  the  two  remainders  and  casting  out 
nines  from  the  product,  we  have  3  for  a  remainder ; 
and  the  remainder  found  by  casting  out  nines  from 
the  product  being  the  same,  we  judge  the  work  to  be 


647)2514157(3885 
1941 
5731 
5176 


8 

48—3 


5555 
5176 

3797 
3235 


562—7 


correct. 

Applied  to  Division. 
6 

Explanation. — Casting  out  the  nines  from 
the  divisor  and.  quotient  separately,  we  find  the 
remainder  written  over  the  last  figure  of  each. 
We  then  multiply  the  one  remainder  by  the 
other,  cast  out  the  nines  of  the  product,  and 
carry  the  excess  to  the  remainder  found  by 
the  division  562,  and  think  3  +  5  +  6  =  14, 
14  —  9  =  5,  and  5  +  2  =  7.  We  finally  cast 
out  the  nines  from  the  dividend,  and  since  we 
obtain  7  from  this  also  we  judge  the  work  to 
be  correct. 


467.  The  principle  on  which  this  method  is  based  is,  that 
The  remainder  arising  from  dividiny  any  number  by  9  is  always 
the  same  as  the  remainder  that  arises  from  dividing  the  sum  of 
all  its  terms  by  9. 

That  this  must  be  so  is  evident  from  the  fact  that,  on  being 
divided  by  9,  there  is  a  remainder  of  1  for  every  ten,  hundred,  or 
thousand  that  go  to  make  up  a  number,  thus  : 

Dividing  10,  100,  or  1000  by  9,  the  remainder  is  always  1.  Dividing  20,  200, 
or  2000  by  9,  the  remainder  is  always  2,  etc.  Hence,  if  we  divide  separately  the 
parts  of  a  number  represented  by  its  digits  by  9,  the  remainders  will  be  expressed 
by  those  digits :  e.  g.,  if  we  divide  the  2000,  400,  70,  and  8,  in  2478,  separately 
by  9,  the  remainders  will  be  2,  4,  7,  and  8,  the  excess  of  9's  in  the  sum  of 
which  is  evidently  the  same  as  in  the  sum  of  the  digits  or  in  the  number  itself. 

468.  Another  excellent  test  of  the  correctness  of  an  opera- 
tion in  division  is  that  the  remainder  after  division  added  to  all 
the  subtrahends  produces  a  sum  equal  to  the  dividend. 


APPENDIX.  409 

Greatest  Common  Divisor.    (See  Art.  141.) 
Example. — Let  it  be  required  to  find  the  greatest  common 
divisor  of  91  and  224. 

The  process  as  given  in  Art.  141,  page  142,  is  as  follows  : 

We  divide  the  greater  number  by  the  less,  and  the 
divisor  by  the  remainder,  and  so  on  till  we  find  that  7  will       91)#24(* 
exactly  divide  the  preceding  divisor  or  remainder,  as  we  182 

choose  to  regard  it.     By  trial,  we  find  that  7  is  a  common  42)91(2 

divisor  of  91  and  224.     But,  by  reasoning,  we  might  con-  "     * 

elude  that  it  must  always  be  the  case  that  the  last  re-  *±±_ 

mainder  in  such  a  succession  of  divisors  will  be  a  common  7)42(6 

divisor  of  the  given  numbers.     The  reasoning  would  be  42 

based  on  two  principles  : 

1.  That  an  exact  divisor  of  any  number  must  be  an  exact  divi- 
sor of  any  multiple  (number  of  times)  that  number. 

If  there  is  an  exact  whole  number  of  times  7  apples  in  one  heap,  there  would 
be  an  exact  whole  number  of  times  7  apples  of  the  same  size  in  any  number  of 
equal  heaps. 

2.  That  an  exact  divisor  of  two  numbers  ivill  be  an  exact  divi- 
sor of  their  sum  or  difference. 

If  there  are  5  times  12  buttons  on  one  string,  and  2  times  12  buttons  on 
another,  there  will  be  an  exact  whole  number  of  times  12  buttons  on  one  string 
more  than  the  other,  and  an  exact  whole  number  of  times  12  buttons  on  both. 

With  these  principles  in  view,  to  show  that  the  last  divisor  is 
a  common  divisor,  we  would  reason  thus  : 

Seven  being  a  divisor  of  42,  it  is,  according  to  principle  1,  a  divisor  of  84  ;  and 
being  a  divisor  of  84,  it  is  (principle  2)  a  divisor  of  91  (84  +  7) ;  and  being  a 
divisor  of  91,  it  is  a  divisor  (principle  1)  of  182  (2  x  91) ;  and  being  a  divisor  of 
182  and  42,  it  is  (principle  2)  a  divisor  of  their  sum,  224.  Hence,  7  must  be  an 
exact  divisor  of  91  and  224. 

And  to  show  that  the  last  divisor  is  the  greatest  common 
divisor,  we  would  reason  as  follows  : 

According  to  principle  1,  the  common  divisor  of  91  and  224  must  be  an  exact 
divisor  of  182  (2  x  91) ;  and  hence,  being  a  common  divisor  of  182  and  224,  it 
must  be,  according  to  principle  2,  an  exact  divisor  of  their  difference,  42  ;  and, 
reasoning  in  a  similar  manner,  being  a  divisor  of  42,  it  must  be  a  divisor  of 
91  —  84,  or  7.     Hence,  the  greatest  common  divisor  can  not  be  greater  than  7. 

Thus,  having  found  that  7  is  a  common  divisor,  and  that  the  common  divisor 
can  not  be  greater  than  7,  we  conclude  that  7  is  the  greatest  common  divisor. 


410  STANDARD  ARITHMETIC. 

Repetends,  or  Circulating  Decimais. 

1.  Decimals,  equivalent  to  the  common  fractions,  given  in 
exercises  10  and  11,  page  149,  were  easily  found,,i)ut  in  12,  the 
division  of  the  numerators  by  the  denominators  was  interminable. 
The  decimals  produced  might  therefore  be  called  interminate 
(not  terminating). 

2.  But,  if  the  division  were  carried  far  enough  (never  to  a  num- 
ber of  places  in  the  quotient  greater  than  the  number  represented 
by  the  divisor),  a  remainder  would  be  obtained  which  had  occurred 
before,  and  hence  a  figure  or  set  of  figures  would  be  repeated  in 
the  same  order  in  never-ending  succession.  Such  a  figure,  or  set 
of  figures,  was  called  a  circulating  decimal  or  repetend. 

3.  Let  the  pupil  now  reduce  y9,  or  any  number  of  9ths  less 
than  9,  to  a  decimal ;  also  y99,  or  any  number  of  99ths  less  than  99, 
to  a  decimal ;  also  y999,  or  any  number  of  999ths  less  than  999,  to 
a  decimal ;  also  any  number  of  9999ths  less  than  9999,  to  a  decimal, 
and  let  him  note  the  several  results.  He  will  in  every  case  obtain 
a  repetend  consisting  of  the  same  digits  as  the  numerator. 

4.  From  these  experiments  it  may  be  safely  concluded  that  any 
repetend  is  equivalent  to  a  common  fraction  having  the  repetend 
for  its  numerator,  and  a  number  of  9's  in  the  denominator  equal 
to  the  number  of  places  in  the  repetend.  Hence  the  method  of  re- 
ducing repetends  to  common  fractions  becomes  evident. 

Example. — l.  Reduce  18  to  a  common  fraction. 

Process. 

18  =  18/99  =  2/n. 

Change  to  common  fractions  : 

2.  .285714  3.  .53846i  4.  .i53846  5.  .045 


6.  Change  .172  to  a  common  fraction. 
.172  =  .17%  = 


1779       155        31 


100        900       180 
Change  to  common  fractions  and  mixed  numbers  ; 
7.  .245  8.  17.53i  9.  .24i62  10.   15.1893 


APPENDIX.  411 

Progression. 

469.  An  Arithmetical  Progression  is  a  series  of  numbers, 
increasing  or  decreasing  by  a  constant  difference. 

1,  3,  5,  7,  9,  11,  is  an  increasing,  or  ascending  series. 

12,  10,  8,  6,  4,  2,  is  a  decreasing,  or  descending  series.  The 
constant  difference  in  each  is  2. 

To  find  any  term  of  an  Arithmetical  Series. 

1.  If  the  first  term  of  an  arithmetical  series  is  2,  and  the  con- 
stant difference  is  3,  what  is  the  fifth  term  ? 

Analysis. — Since  the  constant  difference  is  3,  the  terms  of  the  series  are 
|_2j        1 2+3 1        1 2+3+3]         [2+3+3+3]         [2+3+3+3+3] 

1st  2d  3d  4th  5th 

Here  we  see  that  the  fifth  term  is  equal  to  the  first  term  +  the  constant  differ- 
ence multiplied  by  a  number  one  less  than  the  number  of  the  term  required. 

2  +  (4  X  3)  =  14,  the  fifth  term. 

2.  If  the  first  term  of  a  descending  arithmetical  series  is  30, 
and  the  constant  difference  is  5,  what  is  the  fourth  term  ? 

Analysis. — Since  the  constant  difference  is  5,  the  terms  of  the  series  are 

[30]  [30-5]  1 30-5-5]  [30-5-5-5] 

1st  2d  3d  4th 

Ilere  we  have  the  fourth  term  equal  to  the  first  term  less  the  constant  differ- 
ence multiplied  by  a  number  one  less  than  the  number  of  the  term  required. 

30  —  (3  X  5)  s=  15,  the  fourth  term. 

Hence,  having  the  first  term  and  the  constant  difference,  to  find  any  required 
term  we  have  the 

4*7 0.  Mule* — To  the  first  term  add  the  product  of  the  constant 
difference  by  a  number  one  less  than  the  term  required,  if  the 
series  is  ascending;  or,  if  the  series  is  descending,  subtract  the 
product  from  the  first  term. 

3.  If  the  first  term  of  an  ascending  series  is  6,  the  constant 
difference  3,  and  the  number  of  terms  300,  what  is  the  last 
term  ?     The  seventh  term  ?     The  tenth  term  ? 

4.  The  first  term  of  a  descending  series  is  110,  constant  dif- 
ference 6.     What  is  the  seventh  term  ? 


412  STANDARD  ARITHMETIC. 

5.  If  the  first  term  of  a  descending  series  is  72,  and  the  con- 
stant difference  6,  what  is  the  seventh  term  ?  The  ninth  term  ? 
The  twelfth  term  ? 

6.  A  bicyclist  travels  5  miles  the  first  hour,  and  increases  his 
speed  %  mile  each  hour  for  fifteen  hours.  How  far  does  he  travel 
during  the  last  hour  ? 

To  find  the  sum  of  an  Arithmetical  Series. 

7.  The  first  term  of  an  ascending  series  is  5,  the  constant  dif- 
ference is  3,  and  the  number  of  terms  8.     What  is  the  sum  ? 

The  complete  series     5,         8,       11,       14,       17,       20,       23,       26. 
The  inverted  series   ??        2j        ?0        1_7        14        11  8  5 

31*     3?     31'     31*     31'     31'     31'     31' 

From  this  we  see  that  the  sum  of  the  two  series,  or  twice  the  sum  of  one  series, 
is  equal  to  the  sum  of  the  first  and  last  terms  taken  as  many  times  as  there  are 
terms;  hence  the  sum  of  either  series  is  equal  to  l/2  the  sum  of  the  first  and  last 
terms  taken  as  many  times  as  there  are  terms.  Therefore,  having  the  first  term, 
the  constant  difference,  and  the  number  of  terms,  to  find  the  sum  of  the  terms,  we 
have  the  following 

4-71.  Kiile. — Find  the  last  term,  according  to  the  previous 
rule ;  multiply  the  sum  of  the  first  and  last  terms  by  the  number 
of  terms,  and  divide  the  product  by  2. 

8.  The  extremes  of  a  series  are  3  and  78,  the  number  of  terms 
16.     What  is  the  sum  ? 

9.  How  many  strokes  does  a  clock  make  in  12  hours  ?  How 
many  would  it  make  if  it  struck  the  hours  from  1  to  24. 

10.  To  cancel  a  debt,  A  agrees  to  pay  B  $100  a  year  for  five 
years,  with  an  increase,  after  the  first  year,  of  $5  per  month. 
What  was  the  debt  ? 

11.  The  first  term  of  a  series  is  3,  the  constant  difference  is  5, 
and  the  number  of  terms  99.  What  is  the  twelfth  term  ?  The 
ninety-eighth  term  ?    The  sum  ? 

12.  A  railroad  train  ran  26  miles  the  first  hour,  and  increased 
its  speed  4  miles  each  hour.  How  fast  was  it  running  the  fifth 
hour  after  starting  ?    What  was  the  entire  distance  traveled  ? 


APPENDIX.  413 

472.  It  can  be  readily  seen  that  an  arithmetical  progression 
has  five  elements  which  enter  into  the  solution  of  the  various 
kinds  of  problems  coming  under  this  head,  viz.  :  The  first  term ; 
the  constant  difference  ;  the  number  of  terms  ;  the  last  term  ;  and 
the  sum  of  all  the  terms.  If  any  three  of  these  elements  be  given, 
the  other  two  can  readily  be  found. 

Pupils  should  make  other  problems  and  deduce  rules  for  find- 
ing other  elements  than  those  required  in  the  foregoing  exercises. 


473.  A  Geometrical  Progression  is  a  series  of  numbers 
which  increase  or  decrease  by  a  constant  ratio. 

2,  6,  18,  54,  162  is  a  geometrical  progression,  the  ratio  of 
which  is  3. 

64,  32,  16,  8,  4,  2,  1  is  a  decreasing  geometrical  progression, 
the  ratio  of  which  is  %. 

To  find  any  term  of  a  Geometrical  Series. 

13.  The  first  term  of  a  geometrical  series  is  2,  and  the  ratio  3. 
What  is  the  fourth  term  ? 


Or, 


Thus  we  see  the  fourth  term  is  equal  to  the  first  term  multiplied  by  the  third 
power  of  the  ratio. 

Hence,  for  finding  any  term  of  a  geometrical  series,  we  have  the 

4-74-.  Rule,— Multiply  the  first  term  by  the  ratio  raised  to  a 
power  one  less  than  the  number  of  the  term  required. 

14.  A  father  once  made  the  following  contract  with  his  joking 
son.  For  certain  services  rendered,  he  agreed  to  pay  him  one 
cent  the  first  month,  two  cents  the  second,  four  cents  the  third, 
eight  cents  the  fourth,  and  so  on  at  the  same  rate  for  a  term  of 
five  years.  If  the  contract  could  have  been  carried  out,  what 
would  have  been  the  last  payment  ? 


1*1 

[3X2] 

|3X3X2| 

|3X3X3X2[ 

1st 

2d 

3d 

4th 

l3l 

|3X2| 

|32X2, 

|33X2, 

1st 

2d 

3d 

4th 

414  STANDARD  ARITHMETIC. 

To  find  the  sum  of  a  Geometrical  Series. 

15.  What  is  the  sum  of  a  geometrical  series  whose  first  term  is 
5,  ratio  4,  number  of  terms  6  ? 

The  sum  of  the  series  would  be 

5  +  20  +  80  +  320  +  1280  +  5120. 

If,  now,  we  multiply  this  by  the  ratio  4,  we  have  the  sum  of  a  new  series.  Sub- 
tracting from  this  the  sum  of  the  first  series,  we  have  left  a  result  equal  to  three 
times  the  sum  of  the  first  series,  viz. : 

20   +   $0   +   $U   +  W0   +   20480  Sum  of  new  series. 

5   +   #0   +   $0   +   $20   +  jggjj Sum  of  first  series. 

Result,  20480  —  5  =  to  three  times 

the  sum  of  first  series. 
20475  -r-  3  =  6825  =  Sum  of  first  series. 

Hence,  having  the  first  term,  the  number  of  terms,  and  the  ratio  of  an  increas- 
ing geometrical  series,  to  find  the  sum  of  the  terms  we  have  the 

4-75.  Mule,— Find  the  last  term;  multiply  the  last  term  by 
the  ratio,  from  the  product  subtract  the  first  term,  and  divide  the 
remainder  by  the  ratio  —  1. 

To  find  the  sum  of  a  decreasing  arithmetical  series,  the  following  is  the 

476.  Bide.— "Find  the  last  term;  multiply  the  last  term  by 
the  ratio,  subtract  the  product  from  the  first  term,  and  divide  the 
remainder  by  1  —  the  ratio. 

Thus  it  is  seen  that  in  the  last  rule  the  last  two  steps  are  the  reverse  of  the  last 
two  in  the  first  rule* 

16.  The  first  term  of  a  geometrical  series  is  5,  the  ratio  3, 
number  of  terms  10.     What  is  the  last  term  ? 

17.  What  is  the  sum  of  a  geometrical  series  whose  first  term  is 
5,  ratio  6,  and  number  of  terms  8  ? 

18.  The  first  term  of  a  geometrical  series  is  3125,  the  ratio  is 
y5.     What  is  the  fourteenth  term  ? 

19.  What  is  the  sum  of  the  infinite  series  2,  1,  %  %,  %, 
%  etc.  ? 

Note. — As  the  number  of  terms  in  a  descending  series  increases  the  last  term 
decreases,  and  when  the  number  of  terms  becomes  infinite,  the  last  term  becomes  0. 
Hence  the  sum  =  the  first  term  -f-  (1  —  the  ratio). 


APPENDIX. 


415 


Announced 


Values  of  Foreign  Coins 

the  United  States  Treasury  Department,  January  1,  1886. 


Country. 


Argentine  Republic 

Austria 

Belgium 

Bolivia 

Brazil 

British  possessions  in  N.  A. 

Chili 

Cuba 

Denmark 

Ecuador 

Egypt 

France 

German  Empire 

Great  Britain 

Greece 

Hayti 

India 

Italy 

Japan  

Liberia 

Mexico 

Netherlands 

Norway 

Peru 

Portugal 

Russia 

Spain 

Sweden 

Switzerland 

Tripoli 

Turkey 

United  States  of  Columbia. 
Venezuela 


Monetary  unit. 

Standard. 

Value  in 
U.S.  money. 

Peso 

Gold  and  silver. 
Silver 

96  5 

.37,1 
.19,3 
.75,1 
54  6 

Franc 

Gold  and  silver. 
Silver 

Milreisof  1000  reis... 
Dollar 

Gold 

Gold 

$1.00 
.91,2 

Peso 

Gold  and  silver. 
Gold  and  silver . 
Gold 

Peso 

Crown 

.93,2 

.26,8 

Peso 

Silver 

.75,1 
.04,9 

Piaster  

Gold 

Franc  

Gold  and  silver. 
Gold 

.19,3 

Mark 

.23,8 

Pound  sterling 

Drachma 

Gold 

Gold  and  silver. 
Gold  and  silver. 
Silver 

4.86,6l/2 
.19,3 

Gourde 

Rupee  of  16  annas. . . 

Lira 

Yen 

Dollar 

.96,5 

.35,7 

Gold  and  silver. 
Silver 

.19,3 
.81,0 

Gold 

Silver 

1.00 

Dollar 

Florin 

.81,6 

Gold  and  silver. 
Gold 

.40,2 
.26,8 

Sol : 

Silver 

.75,1 
1.08 

Milreis  of  1000  reis.. . 
Rouble  of  100  copecks. 
Peseta  of  100  centimes. 
Crown , 

Gold 

Silver 

.60,1 

Gold  and  silver. 
Gold 

.19,3 

.26,8 

Franc 

Gold  and  silver. 
Silver 

.19,3 

.67.7 

Mahbub  of  20  piasters. 
Piaster  

Gold 

.04,4 

Peso 

Silver 

Gold  and  silver. 

.75,1 

Bolivar 

.19,3 

Note. — Let  the  student  observe  that  the  monetary  unit  of  the  States  in  the 
northwestern  part  of  Europe  is  the  crown  ;  of  the  States  in  the  south  and  south- 
western part  of  Europe  is  the  franc,  under  different  names ;  and  of  the  north- 
western part  of  South  America  the  peso,  and  he  will  have  no  difficulty  in  remem- 
bering nearly  one  half  of  this  table. 


416 


STANDARD  ARITHMETIC. 


Table  showing   Legal    Rates  of   Interest  in  the  several 

States.     (See  Art.  300,  page  300.) 
[From  "The  Bankers'  Almanac  and  Register,"  1886.] 


States; 

Rate  %. 

Alabama 

8 

, , 

Arizona 

10 

# 

Arkansas 

0 

10 

California 

1 

* 

Colorado 

10 

* 

Connecticut . . . 

6 

6 

Dakota 

n 

12 

Delaware 

6 

6 

Dist.  Columbia. 

6 

10 

Florida 

8 

* 

Georgia 

1 

8 

Idaho 

10 

18 

Illinois 

6 

8 

Indiana 

6 

8 

Iowa 

6 

1 

10 
12 

Kansas 

States. 

Rate  %. 

Kentucky 

6 

6 

Louisiana 

5 

8 

Maine 

6 

* 

Maryland 

6 

6 

Massachusetts . 

6 

* 

Michigan 

1 

10 

Minnesota 

1 

10 

Mississippi. .  .  . 

G 

10 

Missouri 

6 

10 

Montana 

10 

* 

Nebraska 

1 

10 

Nevada  

10 

* 

New  Hampshire 

6 

6 

New  Jersey  . . . 

6 

6 

New  Mexico. . . 

6 

12 

New  York  .... 

6 

6 

States. 

Rate  %. 

North  Carolina. 

6 

8 

Ohio 

6 
8 

8 
10 

Oregon 

Pennsylvania . . 

6 

6 

Rhode  Island 

6 

* 

South  Carolina. 

1 

10 

Tennessee 

6 

6 

Texas 

8 

12 

Utah 

10 

* 

Vermont 

6 

7 

Virginia 

6 

6 

Wash.  Tcr 

10 

# 

West  Virginia. 

6 

6 

Wisconsin  .... 

7 

10 

Wyoming 

12 

* 

The  legal  rate  is  to  be  found  in  the  first  column.     The  second  column  gives  the 
rate  that  may  be  collected  if  agreed  to  in  writing.     *  No  limit. 


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